26. For this problem assume that it is possible to borrow and lend risklessly at a rate of 4%. Also assume that the expected return on the tangency (i.e., the optimal) portfolio composed only of risky assets is 13% with a standard deviation of 18%. Below we list 6 pairs of expected return and standard deviation combinations. For each pair determine whether or not the pair is feasible. If it is feasible, then there is at least one investment that can be made using risky assets and riskless borrowing or lending that produces this level of expected return and standard deviation. Then, if the pair is feasible, determine whether it is efficient or not. It is efficient if the expected return is the highest level that can be obtained for the associated level of standard deviation. Pair Standard Deviation Expected Return a 20.00% 24.75% b 12.00% 18.00% c 30.00% 19.00% d 60.00% 50.00% e 10.00% 4.00% f 45.00% 56.50% 27. Parmacheenee Belle's entire common stock portfolio ($500,000) is allotted to an index fund tracking the Standard & Poors 500 index. The expected rate of return on the index is 9.5% and the standard deviation is 18% per year. The one-year risk-free rate is 2.0%. Now Ms. Belle receives a strongly favorable security analyst's report on Mycronics Corp. The analyst projects a return of 25%. Myroncis has a high volatility (40% annual standard deviation) but its correlation coefficient with the S&P 500 is only .3. Assume the return in the analyst's report is an unbiased forecast. Should Ms. Belle sell part of her index fund holdings and invest in Myronics? If so, how much? Note: Ms. Belle can also lend or borrow at the 2.0% risk-free rate. 28. Samantha Darling's entire common stock portfolio ($100,000) is allotted to an index fund tracking the Standard & Poors 500 index. The 27 c 2008, Andrew W. Lo and Jiang Wang 1.6 Risk & Portfolio Choice 1 QUESTIONS expected rate of return on the index is 12% and the standard deviation is 16% per year. The one-year risk-free rate is 5.5%. Now Samantha receives a strongly favorable security analyst's report on e.Coli Corp. The analyst projects a return for e.Coli of 25%. e.Coli has a high volatility (50% annual standard deviation) but its correlation coefficient with the S&P 500 is only .4. Assume the analyst's report is accurate. Should Samantha sell part of her index fund holdings and invest in e.Coli? If so, now much? Note: Samantha can also lend or borrow at the 5.5% risk-free rate. 29. You are a salesman/investment advisor working for a major investment bank. Whenever clients contact you with money to invest, your job is to help them find an appropriate mutual fund to invest in given their financial position. The available investments are: Fund E[R] (R) A 10% 15% B 20% 45% C 20% 55% Assume throughout this problem that you only recommend one of the three funds to your clients (possibly a different recommendation for different clients though). a) Would you recommend investment C to someone who comes to you with all his investment funds? Explain. b) Which investment would you recommend to Keith Richards, the really, really old rock star from the Rolling Stones? Assume he invests all his wealth in your particular recommendation. c) Might your answer to b) change if Keith invests only half his wealth in your particular recommendation? If so, under what circumstances? 30. Sarah runs an investment consulting business offering advice to clients on portfolio choices, using what she has learned in 15.401. Her analysis shows that the efficient frontier of risky assets can be obtained by mixing two portfolios, a portfolio of "large cap" stocks (L) and a portfolio of "small cap" stocks (S). In addition, she can also invest in riskless T-Bills (F). For a very risk-averse retiree, Sarah has recommended the following portfolio: 70% in F, 20% in L and 10% in S. For a young, less riskaverse executive, however, Sarah recommends only 10% in F and the 28 c 2008, Andrew W. Lo and Jiang Wang 1.6 Risk & Portfolio Choice 1 QUESTIONS rest in the two risky portfolios. Assume that Sarah has chosen the optimal portfolios for both the old retiree and the young executive. What are the weights for the young executive on the "large cap" and "small cap" portfolios, respectively? (Hint: The tangent portfolio should a combination of portfolios L and S.) 31. Which of the following common stock portfolios is best for a conservative, risk-averse investor? Explain briefly. Expected Expected Standard Deviation Return Risk Premium of Return Portfolio A 19% 13% 20% Portfolio B 16% 10% 16% Portfolio C 13% 7% 12.5% Note: the risk premium is calculated by subtracting a 6% Treasury bill rate from the expected rate of return. The investor can also buy Treasury bills. 32. Suppose the overall stock market is divided in four asset classes: largecap growth stocks (LGR, 40% of the market), large-cap income stocks (LINC, 35% of the market), small-cap growth stocks (SMGR, 15% of the market) and small-cap income stocks (SMINC, 10% of the market). Forecasted returns, standard deviations () and correlation coefficients for these asset classes are given on the table below. You can borrow or lend at the risk-free interest rate of 5%. You have $1 million to invest in some combination of the four asset classes. (You can buy index funds or exchange traded portfolios tracking the asset classes.) LGR LINC SMGR SMINC % of market 0.40 0.35 0.15 0.10 r 0.1438 0.1092 0.1329 0.0931 0.28 0.20 0.30 0.22 Correlations: LGR LINC SMGR SMINC LGR 1 0.65 0.70 0.30 LINC 0.65 1 0.40 0.55 SMGR 0.70 0.40 1 0.45 SMINC 0.30 0.55 0.45 1 29 c 2008, Andrew W. Lo and Jiang Wang 1.6 Risk & Portfolio Choice 1 QUESTIONS (a) What are the expected rate of return and standard deviation of the market portfolio? What is the market's Sharpe ratio (the ratio of expected risk premium to standard deviation)? (b) Can you improve the portfolio's Sharpe ratio by investing more in any of the asset classes? (Hint: Analyze a two-asset portfolio, with the market as one asset and a particular asset class as the other. If you sell some of the market portfolio and put the proceeds in that asset class, you end up over-weighting the asset class.) 30 c 2008, Andrew W. Lo and Jiang Wang 1.7 CAPM 1 QUESTIONS 1.7 CAPM 1. What is the beta of a portfolio with E(rp) = 18%, if rf = 6% and E(rM) = 14%? 2. You are a consultant to a large manufacturing corporation that is considering a project with the following net after-tax cash flows (in millions of dollars): Years from Now After-Tax Cash Flow 0 40 1 10 15 The project's beta is 1.8. Assuming that rf = 8% and E(rM) = 16%, what is the net present value of the project? What is the highest possible beta estimate for the project before its NPV becomes negative? 3. Are the following true or false? (a) Stocks with a beta of zero offer an expected rate of return of zero. (b) The CAPM implies that investors require a higher return to hold highly volatile securities. (c) You can construct a portfolio with a beta of 0.75 by investing 0.75 of the investment budget in bills and the remainder in the market portfolio. 4. Assume that the risk-free rate of interest is 6% and the expected rate of return on the market is 16%. A share of stock sells for $50 today. It will pay a dividend of $6 per share at the end of the year. Its beta is 1.2. What do investors expect the stock to sell for at the end of the year? 5. Assume that the risk-free rate of interest is 6% and the expected rate of return on the market is 16%. A stock has an expected rate of return of 4%. What is its beta? Why would anyone consider buying this risky asset which provides an expected return less than the risk-free rate? 6. In 1997 the rate of return on short-term government securities (perceived to be risk-free) was about 5%. Suppose the expected rate of return required by the market for a portfolio with a beta measure of 1 is 12%. According to the capital asset pricing model (security market line): (a) What is the expected rate of return on the market portfolio? (b) What would be the expected rate of return on a stock with = 0? 31 c 2008, Andrew W. Lo and Jiang Wang 1.7 CAPM 1 QUESTIONS (c) Suppose you consider buying a share of stock at $40. The stock is expected to pay $3 dividends next year and you expect it to sell then for $41. The stock risk has been evaluated by = .5. Is the stock overpriced or underpriced? 7. True or False? (a) CAPM says that all risky assets must have positive risk premium. (b) The expected return on an investment with a beta of 2.0 is twice as high as the expected return on the market. (c) If a stock lies below the security market line, it is under valued. 8. If we regress the stocks' average risk premium (return minus the riskfree rate) on their betas, what should be the slope and the intercept according to the CAPM? 9. If we regress a stock's risk premium on the risk premium of the market portfolio, what should be the slope and the intercept according to the CAPM? 10. The risk-free rate is 5%, the expected return on the market portfolio is 14%, and the standard deviation of the return on the market portfolio is 25%. Consider a portfolio with expected return of 16% and assume that it is on the efficient frontier. (a) What is the beta of this portfolio? (b) What is the standard deviation of its return? (c) What is its correlation with the market return? 11. Your future father-in-law is 60 years old. He shows you his portfolio: Assets Holdings Cash $ 50,000 S&P 500 Index Fund 100,000 Analog Devices Inc. 200,000 He asks you to forecast how much the portfolio will be worth in 5 years when he retires. The risk-free rate is 6% per year, the average return on the market portfolio is 12%, the beta of the S&P index is 1.0, and the beta of Analog Devices is 1.5. (a) What is the expected rate of return on the portfolio, assuming the CAPM holds? (b) What is the forecasted portfolio value after 5 years? 32 c 2008, Andrew W. Lo and Jiang Wang 1.7 CAPM 1 QUESTIONS (c) Your future father-in-law is not impressed with this CAPM "theory" since his portfolio has done much better than your forecasted return over the past five years. What would you say about that? 12. Integral Industries, Inc. (III) has three subsidiaries, A, B, and C. You are negotiating to buy subsidiary C. Subsidiary A and B each contribute to 40% of III's market value and have betas of 0.8 and 1.4, respectively. The company as a whole has a beta of 1.0. What is the beta of subsidiary C? If you end up buying it, what would be C's opportunity cost of capital? The current risk-free rate is 6% and the market risk premium is 6%. 13. Stock 1 and 2 have the same beta of 0.8. But stock 1's return has a standard deviation of 40% and stock 2 has a standard deviation of 60%. How would you compare the risk of these two stocks? Which one do you think should have the higher expected returns? Explain briefly. 14. Stock A has a beta of 0.6 and stock B has a beta of 1.2. They both have a return standard deviation of 40% and the market portfolio has a return standard deviation of 25%. What fractions of the total variances of the two stocks' returns are market risks? 15. Five years of monthly risk premiums give the following statistics for Ampersand Electric common stock (risk premium = rate of return - risk-free rate): is 0.4% per month, with a standard error of 1.2% is 1.2, with a standard error of 0.27 R2 is 0.30 is 7.2% per month. (a) What does measure? What role does it play in the CAPM? Does a positive indicate a higher-than-normal expected return? (b) What does R2 measure? Would a higher R2 increase your confidence in the estimated ? Briefly explain your answers. 16. Consider three stocks: Q, R and S. Beta STD (annual) Forecast for Nov 2008 Dividend Stock Price Q 0.45 35% $0.50 $45 R 1.45 40% 0 $75 S -0.20 40% $1.00 $20 33 c 2008, Andrew W. Lo and Jiang Wang 1.7 CAPM 1 QUESTIONS Use a risk-free rate of 2.0% and an expected market return of 9.5%. The market's standard deviation is 18%. Assume that the next dividend will be paid after one year, at t = 1. (a) According to the CAPM, what is the expected rate of return of each stock? (b) What should today's price be for each stock, assuming the CAPM is correct? 17. Assume the Fama-French 3-factor APT holds with the factor risk premiums given in BM Table 8.5, p. 209. What are the expected rates of return for stocks Q, R and S in the previous question? The factor sensitivities are: bmarket bsize bbooktomarket Q 0.45 0.05 0.14 R 1.45 -0.33 -0.22 S -0.20 1.21 0.64 18. It is November, 2007. The following variance-covariance matrix, for the market (S&P 500) and stocks T and U, is based on monthly data from November 2002 to October 2007. Assume T and U are included in the S&P 500. The betas for T and U are T = 0.727 and U = 0.75. S&P500 T U S&P500 0.0256 0.0186 0.0192 T 0.0186 0.1225 0.0262 U 0.0192 0.0262 0.0900 Average monthly risk premiums from 2002 to 2007 were: S&P500 : 1.0% T : 0.6% U : 1.1% Assume the CAPM is correct, and that the expected future market risk premium is 0.6% per month. The risk-free interest rate is 0.3% per month. (a) What were the alpha's for stocks T and U over the last 60 months? (b) What are the expected future rates of return for T and U? 34 c 2008, Andrew W. Lo and Jiang Wang 1.7 CAPM 1 QUESTIONS (c) What are the optimal portfolio weights for the S&P 500, T and U? Explain. 19. CML and SML: Using the properties of the capital market line (CML) and the security market line (SML), determine which of the following scenarios are consistent or inconsistent with the CAPM. Explain your answers. Let A and B denote arbitrary securities while F and M represent the riskless asset and the market portfolio respectively. (a) Scenario I: Security E[R] A 25% 0.8 B 15% 1.2 (b) Scenario II: Security E[R] (R) A 25% 30% M 15% 30% (c) Scenario III: Security E[R] (R) A 25% 55% F 5% 0% M 15% 30% (d) Scenario IV: Security E[R] A 20% 1.5 F 5% 0 M 15% 1.0 (e) Scenario V: Security E[R] A 35% 2.0 M 15% 1.0 20. True/False/Depends Questions: Please include brief explanations in your responses: (a) The average return on stocks in the US over the past 30 years has been 12% annually. You find two portfolios, one with an expected return of 14% and another with an expected return of 19%. This contradicts the CAPM.