Expected Return, Variance, and Covariance 10.1 Ms. Sharp thinks that the distribution of rates of return on Q-mart stock is as follows: State of Economy Depression Recession Normal Boom a. b. Probability of State Occurring 0.1 0.2 0.5 0.2 Q-Mart Stock Return (%) -4.5 4.4 12.0 20.7 What is the expected return on the stock? What is the standard deviation of returns on the stock? 10.2 Suppose you have invested only in two stocks, A and B. The returns on the two stocks depend on the following three states of the economy, which are equally likely to happen: State of Economy Bear Normal Bull Return on Stock A (% ) 6.30 10.50 15.60 Return on Stock B (% ) -3.70 6.40 25.30 a. b. c. 10.3 Calculate the expected return on each stock. Calculate the standard deviation of returns on each stock. Calculate the covariance and correlation between the returns on the two stocks. Mr. Henry can invest in Highbull stock and Slowbear stock. His projection of the returns on these two stocks is as follows: State of Economy Recession Normal Boom Probability of State Occurring 0.25 0.60 0.15 Return on Highbull Stock (% ) -2.00 9.20 15.40 Return on Slowbear Stock (% ) 5.00 6.20 7.40 a. b. c. Portfolios 10.4 Calculate the expected return on each stock. Calculate the standard deviation of returns on each stock. Calculate the covariance and correlation between the returns on the two stocks. A portfolio consists of 120 shares of Atlas stock, which sell for $50 per share, and 150 shares of Babcock stock, which sell for $20 per share. What are the weights of the two stocks in this portfolio? Security F has an expected return of 12 percent and a standard deviation of 9 percent per year. Security G has an expected return of 18 percent and a standard deviation of 25 percent per year. a. What is the expected return on a portfolio composed of 30 percent of Security F and 70 percent of Security G? b. If the correlation between the returns of Security F and Security G is 0.2, what is the standard deviation of the portfolio described in part a? Suppose the expected returns and standard deviations of stocks A and B are E(RA) = 0.15, E(RB) = 0.25, 10.5 10.6 A = 0.1, and B = 0.2, respectively. a. Calculate the expected return and standard deviation of a portfolio that is composed of 40 percent A and 60 percent B when the correlation between the returns on A and B is 0.5. b. Calculate the standard deviation of a portfolio that is composed of 40 percent A and 60 percent B when the correlation coefficient between the returns on A and B is -0.5. c. How does the correlation between the returns on A and B affect the standard deviation of the portfolio? 10.7 Suppose Janet Smith holds 100 shares of Macrosoft stock and 300 shares of Intelligence stock. Macrosoft s stock currently sells at $80 per share, while Intelligence s stock sells at $40 per share. The expected return on Macrosoft s stock is 15 percent, while the expected return on Intelligence s stock is 20 percent. The correlation between the returns on the two stocks is 0.38. a. b. 10.8 Calculate the expected return and standard deviation of her portfolio. Today she sold 200 shares of Intelligence stock in order to pay her tuition. Calculate the expected return and standard deviation of her new portfolio. Consider the possible rates of return on Stocks A and B over the next year: State of Economy Recession Normal Boom Probability of State Occurring 0.2 0.5 0.3 Stock A Return if State Occurs (% ) 7 7 7 Stock B Return if State Occurs (% ) -5 10 25 a. Determine the expected returns, variances, and standard deviations for Stock A and Stock B. b. Determine the covariance and correlation between the returns of Stock A and Stock B. c. Determine the expected return and standard deviation of an equally weighted portfolio of Stock A and Stock B. 10.9 Suppose there are only two stocks in the world: Stock A and Stock B. The expected returns on these two stocks are 10 percent and 20 percent, while the standard deviations of the stocks are 5 percent and 15 percent, respectively. The correlation between the returns on the two stocks is 0. a. Calculate the expected return and standard deviation of a portfolio that is composed of 30 percent A and 70 percent B. b. Calculate the expected return and standard deviation of a portfolio that is composed of 90 percent A and 10 percent B. c. Suppose you are risk averse. Would you hold 100 percent in Stock A? How about 100 percent Stock B? Explain. If a portfolio has a positive weight for each asset, can the expected return on the portfolio be greater than the expected return on the asset in the portfolio with the highest expected return? Can the expected return on the portfolio be less than the expected return on the asset in the portfolio with the lowest expected return? Explain. 10.10 10.11 Miss Maple is considering two securities, A and B, with the relevant information given below: State of Economy Bear Bull a. Probability 0.4 0.6 Return on Security A (%) 3.0 15.0 Return on Security B (%) 6.5 6.5 Calculate the expected return and standard deviation of each of the two securities, b. Suppose Miss Maple invested $2,500 in Security A and $3,500 in Security B. Calculate the expected return and standard deviation of her portfolio. 10.12 10.13 A broker has advised you not to invest in oil industry stocks because they have high standard deviations. Is the broker s advice sound for a risk-averse investor like yourself? Why or why not? There are three securities in the market. The following chart shows their possible payoffs. Probability of Outcome 0.1 0.4 0.4 0.1 Return on Security 1 (%) 0.25 0.20 0.15 0.10 Return on Security 2 (%) 0.25 0.15 0.20 0.10 Return on Security 3 (%) 0.10 0.15 0.20 0.25 State 1 2 3 4 a. b. What is the expected return and standard deviation of each security? What are the covariances and correlations between the pairs of securities? c. What is the expected return and standard deviation of a portfolio with half of its funds invested in Security 1 and half in Security 2? d. What is the expected return and standard deviation of a portfolio with half of its funds invested in Security 1 and half in Security 3? e. What is the expected return and standard deviation of a portfolio with half of its funds invested in Security 2 and half in Security 3? f. What do your answers in parts (a), (c), (d), and (e) imply about diversification? 10.14 The return on Stock A is uncorrelated with the return on Stock B. Stock A has a 40 percent chance of having a return of 15 percent and a 60 percent chance of a return of 10 percent. Stock B has a one-half chance of a 35 percent return and a one-half chance of a -5 percent return. a. Write a list of all of the possible outcomes and their probabilities. b. What is the expected return on a portfolio with 50 percent invested in Stock A and 50 percent invested in Stock B? 10.15 Assume there are N securities in the market. The expected return on every security is 10 percent. All securities also have the same variance of 0.0144. The covariance between any pair of securities is 0.0064. a. What is the expected return and variance of an equally weighted portfolio containing all N securities? Note: the weight of each security in the portfolio is 1/N. b. What will happen to the variance of the portfolio as N approaches infinity? c. 10.16 What characteristics of a security are most important in the determination of the variance of a welldiversified portfolio? Is the following statement true or false? Explain. The most important characteristic in determining the variance of a well-diversified portfolio is the variance of each of the individual stocks. 10.17 10.18 Briefly explain why the covariance of a security with the rest of a well-diversified portfolio is a more appropriate measure of the risk of the security than the security s variance. Consider the following quotation from a leading investment manager: The shares of Southern Co. have traded close to $12 for most of the past three years. Since Southern s stock has demonstrated very little price movement, the stock has a low beta. Texas Instruments, on the other hand, has traded as high as $150 and as low as its current $75. Since TI s stock has demonstrated a large amount of price movement, the stock has a very high beta. Do you agree with this analysis? Explain. 10.19 The market portfolio has an expected return of 12 percent and a standard deviation of 10 percent. The riskfree rate is 5 percent. a. b. 10.20 What is the expected return on a well-diversified portfolio with a standard deviation of 7 percent? What is the standard deviation of a well-diversified portfolio with an expected return of 20 percent? Consider the following information on the returns on the market and Fuji stock. Type of Economy Bear Bull Return on Market (% ) 2.50 16.3 Expected Return on Fuji (% ) 3.40 12.8 Calculate the beta of Fuji. CAPM 10.21 10.22 William Shakespeare s character Polonius in Hamlet says, Neither a borrower nor a lender be. Under the assumptions of the Capital <a href="/keyword/asset-pricing-model/" >asset pricing model</a> , what would be the composition of Polonius s portfolio? Securities A, B, and C have the following characteristics: Security A B C a. b. c. 10.23 Expected Return (%) 10 14 20 Beta 0.7 1.2 1.8 What is the expected return on a portfolio with equal weights? What is the beta of a portfolio with equal weights? Are the three securities priced correctly according to the Capital <a href="/keyword/asset-pricing-model/" >asset pricing model</a> ? Holup, Inc., makes pneumatic equipment. The beta of Holup s stock is 1.2. The expected <a href="/keyword/market-risk-premium/" >market risk premium</a> is 8.5 percent, and the current risk-free rate is 6 percent. Assume the Capital <a href="/keyword/asset-pricing-model/" >asset pricing model</a> holds. What is the expected return on Holup s stock? The beta of Stock A is 0.80. The risk-free rate is 6 percent, and the <a href="/keyword/market-risk-premium/" >market risk premium</a> is 8.5 percent. Assume the Capital <a href="/keyword/asset-pricing-model/" >asset pricing model</a> holds. What is the expected return on Stock A? The risk-free rate is 8 percent. The beta of Stock B is 1.5, and the expected return on the market portfolio is 15 percent. Assume the Capital <a href="/keyword/asset-pricing-model/" >asset pricing model</a> holds. What is the expected return on Stock B? Suppose the expected <a href="/keyword/market-risk-premium/" >market risk premium</a> is 7.5 percent and the risk-free rate is 3.7 percent. The expected return on TriStar s stock is 14.2 percent. Assume the Capital <a href="/keyword/asset-pricing-model/" >asset pricing model</a> holds. What is the beta of TriStar s stock? Consider the following two stocks: 10.24 10.25 10.26 10.27 Murck Pharmaceutical Pizer Drug Corp Beta 1.4 0.7 Expected Return 25% 14% Assume the Capital <a href="/keyword/asset-pricing-model/" >asset pricing model</a> holds. Based on the CAPM, what is the risk-free rate? What is the expected return on the market portfolio? 10.28 Suppose you observe the following situation: Return if State Occurs Stock A Stock B -0.1 -0.3 0.1 0.05 0.2 0.4 State of Economy Bust Normal Boom Probability of State 0.25 0.5 0.25 a. Calculate the expected return on each stock. b. Assuming the Capital <a href="/keyword/asset-pricing-model/" >asset pricing model</a> holds and Stock A s beta is greater than Stock B s beta by 0.25, what is the expected <a href="/keyword/market-risk-premium/" >market risk premium</a> ? 10.29 Assume the Capital <a href="/keyword/asset-pricing-model/" >asset pricing model</a> holds. a. Draw the security market line for the case where the expected <a href="/keyword/market-risk-premium/" >market risk premium</a> is 5 percent and the risk-free rate is 7 percent. b. Suppose that an asset has a beta of 0.8 and an expected return of 9 percent. Does the expected return of this asset lie above or below the security market line that you drew in part a? Is the security properly priced? If not, explain what will happen in this market. c. Suppose that an asset has a beta of 3 and an expected return of 25 percent. Does the expected return of this asset lie above or below the security market line that you drew in part a? Is the security properly priced? If not, explain what will happen in this market. 10.30 A stock has a beta of 1.8. A security analyst who specializes in studying this stock expects its return to be 18 percent. Suppose the risk-free rate is 5 percent and the expected <a href="/keyword/market-risk-premium/" >market risk premium</a> is 8 percent. Is the analyst pessimistic or optimistic about this stock relative to the market s expectations? Suppose the expected return on the market portfolio is 13.8 percent and the risk-free rate is 6.4 percent. Solomon Inc. stock has a beta of 1.2. Assume the Capital <a href="/keyword/asset-pricing-model/" >asset pricing model</a> holds. a. b. 10.32 What is the expected return on Solomon s stock? If the risk-free rate decreases to 3.5 percent, what is the expected return on the Solomon s stock? 10.31 A portfolio that combines the risk-free asset and the market portfolio has an expected return of 25 percent and a standard deviation of 4%. The risk-free rate is 5 percent, and the expected return on the market portfolio is 20 percent. Assume the Capital <a href="/keyword/asset-pricing-model/" >asset pricing model</a> holds. What expected rate of return would a security earn if it had a 0.5 correlation with the market portfolio and a standard deviation of 2 percent? The risk-free rate is 7.6 percent. Potpourri Inc. stock has a beta of 1.7 and an expected return of 16.7 percent. Assume the Capital <a href="/keyword/asset-pricing-model/" >asset pricing model</a> holds. a. b. c. What is the expected <a href="/keyword/market-risk-premium/" >market risk premium</a> ? Magnolia Industries stock has a beta of 0.8. What is the expected return on the Magnolia stock? Suppose you have invested $10,000 in a combination of Potpourri and Magnolia stock. The beta of the portfolio is 1.07. How much did you invest in each stock? What is the expected return of the portfolio? 10.33 10.34 Suppose the risk-free rate is 6.3 percent and the market portfolio has an expected return of 14.8 percent. The market portfolio has a variance of 0.0121. Portfolio Z has a correlation coefficient with the market of 0.45 and a variance of 0.0169. According to the Capital <a href="/keyword/asset-pricing-model/" >asset pricing model</a> , what is the expected return on Portfolio Z? You have access to the following data concerning the Durham Company and the market portfolio: Variance of returns on the market portfolio = 0.04326 Covariance between the returns on Durham and the market portfolio = 0.0635 The expected <a href="/keyword/market-risk-premium/" >market risk premium</a> is 9.4 percent and the expected return on Treasury bills is 4.9 percent. a. Write the equation of the security market line. b. What is the required return on Durham Company s stock? 10.35 10.36 Johnson Paint stock has an expected return of 19 percent and a beta of 1.7, while Williamson Tire stock has an expected return of 14 percent and a beta of 1.2. Assume the Capital <a href="/keyword/asset-pricing-model/" >asset pricing model</a> holds. What is the expected return on the market? What is the risk-free rate? Is the following statement true or false? Explain. A risky security cannot have an expected return that is less than the risk-free rate because no riskaverse investor would be willing to hold this asset in equilibrium. 10.37 10.38 Suppose you have invested $30,000 in the following four stocks: Security Stock A Stock B Stock C Stock D Amount Invested $5,000 10,000 8,000 7,000 Beta 0.75 1.1 1.36 1.88 The risk-free rate is 4 percent and the expected return on the market portfolio is 15 percent. Based on the Capital <a href="/keyword/asset-pricing-model/" >asset pricing model</a> , what is the expected return on the above portfolio? 10.39 You have been provided the following data on the securities of three firms, the market portfolio, and the risk-free asset: Security Firm A Firm B Firm C The market portfolio The risk-free asset *with the market portfolio Expected Return 0.13 0.16 0.25 0.15 0.05 Standard Deviation 0.12 (ii) 0.24 0.1 (vi) Correlation* (i) 0.4 0.75 (iv) (vii) Beta 0.9 1.1 (iii) (v) (viii) a. b. Fill in the missing values in the table. Is the stock of Firm A correctly priced according to the Capital <a href="/keyword/asset-pricing-model/" >asset pricing model</a> (CAPM)? What about the stock of Firm B? Firm C? If these securities are not correctly priced, what is your investment recommendation for someone with a well-diversified portfolio? 10.40 There are two stocks in the market, Stock A and Stock B. The price of Stock A today is $50. The price of Stock A next year will be $40 if the economy is in a recession, $55 if the economy is normal, and $60 if the economy is expanding. The probabilities of recession, normal times, and expansion are 0.1, 0.8, and 0.1, respectively. Stock A pays no dividends and has a correlation of 0.8 with the market portfolio. Stock B has an expected return of 9%, a standard deviation of 12%, a correlation with the market portfolio of 0.2, and a correlation with Stock A of 0.6. The market portfolio has a standard deviation of 10%. Assume the CAPM holds. a. If you are a typical, risk-averse investor with a well-diversified portfolio, which stock would you prefer? Why? b. What are the expected return and standard deviation of a portfolio consisting of 70 percent of Stock A and 30 percent of Stock B? c. What is the beta of the portfolio in part (b)? Advanced (Requires Calculus) 10.41 Assume stocks A and B have the following characteristics: Stock A B Expected Return (%) Standard Deviation % 5 10 10 20 The covariance between the returns on the two stocks is 0.001. a. b. c. d. Suppose an investor holds a portfolio consisting of only Stock A and Stock B. Find the portfolio weights, XA and XB, such that the variance of his portfolio is minimized. (Hint: Remember that the sum of the two weights must equal 1.) What is the expected return on the minimum variance portfolio? If the covariance between the returns on the two stocks is 0.02, what are the minimum variance weights? What is the variance of the portfolio in part c?