JEM034 Corporate Finance Winter Semester 2017/2018 Instructor: Olga Bychkova Homework #4 Suggested Solutions Problem 1. (7.2) The following table shows the nominal returns on the U.S. stocks and the rate of inflation. (a) What was the standard deviation of the nominal market returns? (b) Calculate the average real return. (a) The following table shows the calculation of the standard deviation of the market returns: Deviation from Squared Deviation from Year Nominal Return (%) Average Nominal Return Average Nominal Return 2004 12.5 11.88 141.1344 2005 6.4 5.78 33.4084 2006 15.8 15.18 230.4324 2007 5.6 4.98 24.8004 2008 37.2 37.82 1430.3524 Average 0.62 372.0256 Standard deviation = 372.0256 = 19.3%. (b) The following table shows the calculation of the real returns: Year Nominal Return (%) Inflation (%) Real Return (%) 2004 12.5 3.3 8.91 2005 6.4 3.4 2.9 2006 15.8 2.5 12.98 2007 5.6 4.1 1.44 2008 37.2 0.1 37.26 Average 2.2 1
where r real = 1 + r nominal 1 + r inflation 1. Problem 2. (7.3) During the boom years of 2003 2007, ace mutual fund manager Diana Sauros produced the following percentage rates of return. Rates of return on the market are given for comparison. Calculate the average return and standard deviation of Ms. Sauros s mutual fund. Did she do better or worse than the market by these measures? The following table shows the calculations of the average returns and standard deviations of Ms. Sauros s and market returns: Ms. Sauros S&P 500 Ms. Sauros Squared S&P 500 Squared Ms. Sauros S&P 500 Deviation from Deviation from Deviation from Deviation from Year Return Return Average Return Average Return Average Return Average Return 2003 39.1 31.6 24.5 600.25 17.22 296.5284 2004 11 12.5 3.6 12.96 1.88 3.5344 2005 2.6 6.4 12 144 7.98 63.6804 2006 18 15.8 3.4 11.56 1.42 2.0164 2007 2.3 5.6 12.3 151.29 8.78 77.0884 Average 14.6 14.38 184.012 88.5696 Ms. Sauros had a slightly higher average return (14.6% vs. 14.38% for the market). However, the fund also had a higher standard deviation (13.6% vs. 9.4% for the market). Problem 3. (7.10) Here are inflation rates and U.S. stock market and Treasury bill returns between 1929 and 1933: (a) What was the real return on the stock market in each year? (b) What was the average real return? (c) What was the risk premium in each year? (d) What was the average risk premium? (e) What was the standard deviation of the risk premium? 2
The following table shows the calculations of the real returns, risk premiums and its standard deviations: Stock Deviation Squared Deviation Market T Bill Real Stock Risk from Average from Average Year Inflation Return Return Market Return Premium Risk Premium Risk Premium 1929 0.2 14.5 4.8 14.23 19.3 9.54 91.0116 1930 6 28.3 2.4 23.72 30.7 20.94 438.4836 1931 9.5 43.9 1.1 38.01 45 35.24 1241.8576 1932 10.3 9.9 1 0.446 10.9 1.14 1.2996 1933 0.5 57.3 0.3 56.52 57 66.76 4456.8976 Average 3.8 9.76 1245.91 Standard deviation = 1245.91 = 35.3%. Problem 4. (7.14) Hyacinth Macaw invests 60% of her funds in stock I and the balance in stock J. The standard deviation of returns on I is 10%, and on J it is 20%. Calculate the variance of portfolio returns, assuming (a) The correlation between the returns is 1. (b) The correlation is 0.5. (c) The correlation is 0. (a) σ 2 p = x 2 I σ2 I +x2 J σ2 J +2x Ix J ρ IJ σ I σ J = 0.6 2 0.1 2 +0.4 2 0.2 2 +2 0.6 0.4 1 0.1 0.2 = 0.0196 (b) σ 2 p = x 2 I σ2 I +x2 J σ2 J +2x Ix J ρ IJ σ I σ J = 0.6 2 0.1 2 +0.4 2 0.2 2 +2 0.6 0.4 0.5 0.1 0.2 = 0.0148 (c) σ 2 p = x 2 I σ2 I +x2 J σ2 J +2x Ix J ρ IJ σ I σ J = 0.6 2 0.1 2 +0.4 2 0.2 2 +2 0.6 0.4 0 0.1 0.2 = 0.01 Problem 5. (7.22) Suppose that Treasury bills offer a return of about 6% and the expected market risk premium is 8.5%. The standard deviation of Treasury bill returns is zero and the standard deviation of market returns is 20%. Use the formula for portfolio risk to calculate the standard deviation of portfolios with different proportions in Treasury bills and the market. (Note: The covariance of two rates of return must be zero when the standard deviation of one return is zero.) Graph the relationship between the expected returns and standard deviations. For a two security portfolio, the formula for portfolio risk is: P ortfolio variance = x 2 1σ 2 1 + x 2 2σ 2 2 + 2x 1 x 2 ρ 12 σ 1 σ 2. If security one is Treasury bills and security two is the market portfolio, then σ 1 is zero, σ 2 is 20%. Therefore: P ortfolio variance = x 2 2σ 2 2 = x 2 2 0.2 2 Standard deviation = 0.2x 2. 3
Portfolio x 1 x 2 Expected Return Standard Deviation 1 1 0 0.06 0 2 0.8 0.2 0.077 0.04 3 0.6 0.4 0.094 0.08 4 0.4 0.6 0.111 0.12 5 0.2 0.8 0.128 0.16 6 0 1 0.145 0.2 P ortfolio expected return = x 1 0.06 + x 2 (0.06 + 0.085) = 0.06x 1 + 0.145x 2. Problem 6. (8.8) Consider a three factor APT model. The factors and associated risk premiums are Calculate expected rates of return on the following stocks. The risk free interest rate is 7%. (a) A stock whose return is uncorrelated with all three factors (i.e., with b = 0 for each). (b) A stock with average exposure to each factor (i.e., with b = 1 for each). (c) A pure play energy stock with high exposure to the energy factor (b = 2) but zero exposure to the other two factors. (d) An aluminum company stock with average sensitivity to changes in interest rates and GNP, but negative exposure of b = 1.5 to the energy factor. (The aluminum company is energy intensive and suffers when energy prices rise.) r = r f +b GNP r GNP factor +b energy prices r energy prices factor +b long term interest rates r long term interest rates factor. 4
(a) r = 7 + 0 5 + 0 ( 1) + 0 2 = 7%. (b) r = 7 + 1 5 + 1 ( 1) + 1 2 = 13%. (c) r = 7 + 0 5 + 2 ( 1) + 0 2 = 5%. (d) r = 7 + 1 5 + ( 1.5) ( 1) + 1 2 = 15.5%. Problem 7. (8.15) The Treasury bill rate is 4%, and the expected return on the market portfolio is 12%. Using the capital asset pricing model: (a) Draw a graph showing how the expected return varies with beta. (b) What is the risk premium on the market? (c) What is the required return on an investment with a beta of 1.5? (d) If an investment with a beta of 0.8 offers an expected return of 9.8%, does it have a positive NPV? (e) If the market expects a return of 11.2% from stock X, what is its beta? (a) r = r f + β(r m r f ). (b) Market risk premium = r m r f = 12 4 = 8%. (c) Use the security market line: r = r f + β(r m r f ) = 4 + 1.5 (12 4) = 16%. (d) For any investment, we can find the opportunity cost of capital using the security market line. With β = 0.8, the opportunity cost of capital is: r = r f + β(r m r f ) = 4 + 0.8 (12 4) = 10.4%. The opportunity cost of capital is 10.4% and the investment is expected to earn 9.8%. Therefore, the investment has a negative NPV. (e) Again, we use the security market line: r = r f + β(r m r f ), 11.2 = 4 + β(12 4) β = 0.9. Problem 8. (8.18) Some true or false questions about the APT: 5
(a) The APT factors cannot reflect diversifiable risks. (b) The market rate of return cannot be an APT factor. (c) There is no theory that specifically identifies the APT factors. (d) The APT model could be true but not very useful, for example, if the relevant factors change unpredictably. (a) True. By definition, the factors represent macroeconomic risks that cannot be eliminated by diversification. (b) False. The APT does not specify the factors. (c) True. Different researchers have proposed and empirically investigated different factors, but there is no widely accepted theory as to what these factors should be. (d) True. To be useful, we must be able to estimate the relevant parameters. If this is impossible, for whatever reason, the model itself will be of theoretical interest only. Problem 9. (8.19) Consider the following simplified APT model: Calculate the expected return for the following stocks. Assume r f = 5%. r = r f + b 1 r 1 + b 2 r 2 + b 3 r 3. Stock P : r = 5% + 1 6.4% + ( 2) ( 0.6%) + ( 0.2) 5.1% = 11.58%. Stock P 2 : r = 5% + 1.2 6.4% + 0 ( 0.6%) + 0.3 5.1% = 14.21%. Stock P 3 : r = 5% + 0.3 6.4% + 0.5 ( 0.6%) + 1 5.1% = 11.72%. Problem 10. (8.20) Look again at the previous problem. Consider a portfolio with equal investments in stocks P, P 2, and P 3. (a) What are the factor risk exposures for the portfolio? (b) What is the portfolio s expected return? 6
(a) Factor risk exposures: b 1 (Market) = 1 /3 1 + 1 /3 1.2 + 1 /3 0.3 = 0.83, b 2 (Interest rate) = 1 /3 ( 2) + 1 /3 0 + 1 /3 0.5 = 0.5, b 3 (Y ield spread) = 1 /3 ( 0.2) + 1 /3 0.3 + 1 /3 1. = 0.37. (b) r P = 5% + 0.83 6.4% + ( 0.5) ( 0.6%) + 0.37 5.1% = 12.5%. Problem 11. (8.21) The following table shows the sensitivity of four stocks to the three Fama French factors. Estimate the expected return on each stock assuming that the interest rate is 0.2%, the expected risk premium on the market is 7%, the expected risk premium on the size factor is 3.6%, and the expected risk premium on the book to market factor is 5.2%. r Boeing = 0.2% + 0.66 7% + 1.19 3.6% + ( 0.76) 5.2% = 5.152% r J&J = 0.2% + 0.54 7% + ( 0.58) 3.6% + 0.19 5.2% = 2.88% r Dow Chemical = 0.2% + 1.05 7% + ( 0.15) 3.6% + 0.77 5.2% = 11.014% r Microsoft = 0.2% + 0.91 7% + ( 0.04) 3.6% + ( 0.4) 5.2% = 4.346% 7