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21. There are three assets, A, B and C, where A is the market portfolio and C is the risk-free asset. The return on the market has a mean of 12% and a standard deviation of 20%. The risk-free asset yields a return of 4%. Asset B is a risky asset whose return has a standard deviation of 40% and a market beta of 1. Assume that the CAPM holds. (a) Compute the expected return of asset B and its covariances with asset A (the market portfolio) and asset C' (the risk-free asset), respectively. (b) Consider a portfolio of the two risky assets, A and B, with weight w in asset A (the market portfolio) and 1-w in asset B. Compute the expected return and return standard deviation of the portfolio with w being 0, 1/2, and 1, respectively, and enter them into the following table: Portfolio weight w 1/2 Expected return Standard deviation (c) Can you rank the three portfolios in the question above? Explain. (d) Consider a portfolio with equal weights in asset B and C (the risk- free asset). Denote this portfolio as asset D. Compute the return standard deviation and expected return of asset D. (e) Consider a portfolio of asset A (the market portfolio) and C. Find the portfolio weight such that its return standard deviation is the same as that of asset D in Question (d). What is the expected return of this portfolio? (f) What can you say about the mean-variance efficiency of asset A, B and C (i.e., are they efficient portfolios)? Explain. (g) Construct an efficient portfolio from the assets A, B and C with an expected return of 10%. 22. Assume that an investor must put all of her money in the following four stocks. Expected Return Standard Deviation Correlations B C D 11% 25% 1.0 15 14.5% 35% 3 10 .2 9% 30% .15 .45 10 .25 11.5% 32% 2 25 10 25 20106, Andrew W. La and Jiang Wang 1.6 Risk & Portfolio Choice 1 QUESTIONS (a) What is the expected return and standard duration of an equally weighted portfolio of the four stocks? (b) Calculate the mean-variance efficient portfolios that can be con- structed from the four stocks. (Hint: use Excel Solver) (c) Assume the investor can borrow or lend at a 5% risk-free rate. What is the best portfolio of the four stocks? 23. Assets X. Y, and Z have the following characteristics: Asset Exp. Ret. (F) Std. Dev. (o) X 15% 20% 10% 15%% Z 10% 15% Correlation X Y Z X 1 0.7 0.3 0.2 Consider the portfolio frontier of the three assets. Without solving for the portfolio weights, answer the following questions:17. Calculate the expected return and standard deviation of a portfolio of stocks A, B and C. Assume an equal investment in each stock. Expected Return Standard Deviation Correlations A B C A 11% 30% 1.0 .3 .15 14.5% 45% .3 1.0 .45 9% 30% 45 1.0 18. Your employer offers two funds for your pension plan, a money market fund and an S&P 500 index fund. The money market fund holds 3- month Treasury bills, which currently offer a 3% safe return per year. The S&P 500 index fund offers an expected return of 10% per year with a standard deviation of 20%. (a) You want to achieve an expected return of 8% per year for your portfolio. What should be the composition of your portfolio? What is the standard deviation of its returns? (b) Now your employer adds an emerging-market fund to the two ex- isting funds. The emerging-market fund offers an expected return of 10% per year, the same as the S&P 500 index fund, but with a standard deviation of 30%, higher than the S&P 500 index fund. Would you consider including the emerging-market fund as part of your portfolio? Explain. (e) The correlation between the S&P 500 index fund and the emerging- market fund is zero. Consider portfolio A, which consists of 80% in the S&P index fund and 20% in the emerging-market fund. Calculate portfolio A's expected return and standard deviation. (d) If you mix portfolio A with the money-market fund to achieve an expected return of 8%, is it better than the portfolio in part (a) of this question? Explain. 19. You are given data on the following stocks: Stock | E[R] + VIR] Price Mkt. Cap. A 0.10 0.0625 0.0437 0.0525 $50 $105M B 0.15 0.1225 0.0420 $30 $40M 0.20 0.0900 $27 $75M 23 @) 2005, Andrew W. Lo and Jiang Wang 1.6 Risk & Portfolio Choice 1 QUESTIONS a) Compute the expected return and variance of an equally weighted portfolio (i.e. weights of 1/3 on each of the stocks). Consider alternative portfolios formed using assets A, B, and C. For instance, a value-weighted portfolio places weights on each assets in proportion to their market capitalizations. The S&P500 index is an ex- ample of a value-weighted portfolio. A price-weighted portfolio places weights in proportion to prices. The Dow Jones is an example of a price-weighted portfolio. b) What are the expected returns and variance of a value-weighted portfolio. c) What are the expected returns and variance of a price-weighted portfolio. 20. Your current portfolio consists of three assets, the common stock of IBM and GM combined with an investment in the riskless asset. You know the following about the stocks (pay denotes the correlation be- tween asset i and asset j, and M denotes the market portfolio): PIBM,M = 0.60 PGM.M = 0.80 "By = 0.0900 zy = 0.062512. You can form a portfolio of two assets, A and B, whose returns have the following characteristics: Stock | E[R] Standard Deviation Correlation A 0.10 0.20 0.5 B 0.15 0.40 If you demand an expected return of 12%, what are the portfolio weights? What is the portfolio's standard deviation? 13. Your have decided to invest all your wealth in two mutual funds: A and B. Their returns are characterized as follows: . the mean returns are FA = 20% and FB = 15% . the covariance matrix is TA TB TA 0.3600 0.0840 TB 0.0840 0.1225 21 @ 2001, Andrew W. Lo and Jiang Wang 1.6 Risk & Portfolio Choice 1 QUESTIONS You want your total portfolio to yield a return of 18%. What proportion of your wealth should you invest in fund A and B? What is the standard deviation of the return on your portfolio? 14. In addition to the fund A and B in the previous question, now you decide to include fund C to your portfolio. Its expected return is fc = 10%. The covariance matrix of the three funds is TA TB TC TA 0.3600 0.0840 0.1050 TB 0.0840 0.1225 0.0700 TC 0.1050 0.0700 0.0625 Your portfolio now consists of fund A, B and C. You would like to have an expected return of 16% on your portfolio and a minimum risk (measured by standard deviation of the return). What portfolio should you hold? What is the return standard deviation of your portfolio? (Hint: You would need to use Excel Solver or some other optimization software to solve the optimal portfolio.) 15. You can only invest in two securities: ABC and XYZ. The correlation between the returns of ABC and XYZ is 0.2. Expected returns and standard deviations are as follows: Security | E[R] (R) ABC 20% 20% XYZ 15% 25% a) It seems that ABC dominates XYZ in that it has a higher expected return and lower standard deviation. Would anyone ever invest in XYZ? Why? b) What is the expected return and standard deviation of a portfo- lio that invests 60% in ABC and 40% in XYZ? c) Suppose instead that you want your portfolio to have an expected return of 19.5%. What portfolio weights do you select now? What is the standard deviation of this portfolio? 16. You have the same data as the previous question. In addition, you have a risk-free security with a guaranteed return of 5%. The tangency portfolio has an expected return of ?? and standard deviation of ??. (a) What weights are placed on ABC and XYZ in the tangency port- folio?12. You can form a portfolio of two assets, A and B, whose returns have the following characteristics: Stock | E[R] Standard Deviation Correlation A 0.10 0.20 0.5 B 0.15 0.40 If you demand an expected return of 12%, what are the portfolio weights? What is the portfolio's standard deviation? 13. Your have decided to invest all your wealth in two mutual funds: A and B. Their returns are characterized as follows: . the mean returns are FA = 20% and FB = 15% . the covariance matrix is TA TB TA 0.3600 0.0840 TB 0.0840 0.1225 21 @ 2001, Andrew W. Lo and Jiang Wang 1.6 Risk & Portfolio Choice 1 QUESTIONS You want your total portfolio to yield a return of 18%. What proportion of your wealth should you invest in fund A and B? What is the standard deviation of the return on your portfolio? 14. In addition to the fund A and B in the previous question, now you decide to include fund C to your portfolio. Its expected return is fc = 10%. The covariance matrix of the three funds is TA TB TC TA 0.3600 0.0840 0.1050 TB 0.0840 0.1225 0.0700 TC 0.1050 0.0700 0.0625 Your portfolio now consists of fund A, B and C. You would like to have an expected return of 16% on your portfolio and a minimum risk (measured by standard deviation of the return). What portfolio should you hold? What is the return standard deviation of your portfolio? (Hint: You would need to use Excel Solver or some other optimization software to solve the optimal portfolio.) 15. You can only invest in two securities: ABC and XYZ. The correlation between the returns of ABC and XYZ is 0.2. Expected returns and standard deviations are as follows: Security | E[R] (R) ABC 20% 20% XYZ 15% 25% a) It seems that ABC dominates XYZ in that it has a higher expected return and lower standard deviation. Would anyone ever invest in XYZ? Why? b) What is the expected return and standard deviation of a portfo- lio that invests 60% in ABC and 40% in XYZ? c) Suppose instead that you want your portfolio to have an expected return of 19.5%. What portfolio weights do you select now? What is the standard deviation of this portfolio? 16. You have the same data as the previous question. In addition, you have a risk-free security with a guaranteed return of 5%. The tangency portfolio has an expected return of ?? and standard deviation of ??. (a) What weights are placed on ABC and XYZ in the tangency port- folio?8. Which of the following portfolios can not be on the Markowitz efficient frontier? Explain briefly. Portfolio Expected Return Standard Deviation 10% 15% 10.5% 16.5% 11.5% 18.5% 12.5% 20% 9. You have decided to invest all your wealth in two mutual funds: A and B. Their returns and risks are as follows: . the mean returns are FA = 15% and FB = 11% . the covariance matrix is TA TA .04 025 TB .025 032 You want your total portfolio to yield a return of 12%. What pro- portions of your wealth should you invest in A and B? What is the standard deviation of the return on your portfolio? 10. There are only two securities (A and B, no risk free asset) in the market. Expected returns and standard deviations are as follows: Security| Expected return | standard Deviation Stock A 25% 20%% Stock B 15% 25% (a) The correlation between stocks A and B is 0.8. Compute the expected return and standard deviation of a portfolio that has 0% of A, 10% of A, 20% of A, etc, until 100% of A. Plot the portfolio frontier formed by these portfolios. 20 2004, Andrew W. La and Jiang Wang 1.6 Risk & Portfolio Choice 1 QUESTIONS (b) Repeat the previous question, assuming that the correlation is -0.8. (c) Explain intuitively why the portfolio frontier is different in the two cases. 11. Stock A and B have the following characteristics: E(r) 8% 20% 8% 40% Their correlation is 0. The risk-free interest rate is 2%. (a) Consider a portfolio, P, with 90% in stock A and 10% in the risk- free asset. What is the mean and standard deviation of portfolio P's return? (b) Consider another portfolio, Q, which consists of 80% of stock A and 20% of stock B. What is the mean and standard deviation of portfolio Q's return? (c) You need to choose a portfolio to invest all your wealth in. Be- tween portfolio P and Q, which one is better? Explain why. (d) Given that stock A dominates stock B (A has the same mean but lower risk), explain why you ever include stock B in your portfolio. 12. You can form a portfolio of two assets, A and B, whose returns have the following characteristics:1.6 Risk & Portfolio Choice 1. True or false or "it depends"? (a) Briefly explain or qualify your answer: diversification can reduce risk only when asset returns are negatively correlated. (b) If the returns on all risky assets in the world were uncorrelated with each other, the expected return of each risky asset should be the same. 2. True or false or "it depends"? Optimal portfolios should exclude indi- vidual assets whose expected return and risk (measured by its standard deviation) are dominated by other available assets. 3. Is the following statement true or false? Explain. As more securities are added to a portfolio, total risk would typically be expected to fall at a decreasing rate. 4. You need to invest $10M in two assets: a risk-free asset with an ex- pected return of 5% and a risky asset with an expected return of 12% and a standard deviation of 40%. You face a cap of 30% on the port- folio's standard deviation (the "risk budget"). What is the maximum expected return you can achieve on your portfolio? 5. Are the following statements true, false or uncertain? Briefly explain your answer. (a) Diversification over a large number of assets completely eliminates risk. (b) Diversification works only when asset returns are uncorrelated. (c) A stock with high standard deviation may contribute less to port- folio risk than a stock with lower standard deviation. (d) Diversification reduces the expected return on the portfolio as its risk decreases. 6. Are the following statements true or false? Give brief but precise ex- planations for your answers. (a) Stock A has expected return 10% and standard deviation 15%, and stock B has expected return 12% and standard deviation 13%. Then, no investor will buy stock A. (b) Diversification means that the equally weighted portfolio is opti- mal. 7. Which statement about portfolio diversification is correct? 19 5 2001, Andrew W. Lo and Jiang Wang 1.6 Risk & Portfolio Choice 1 QUESTIONS (a) Proper diversification can reduce or eliminate systematic risk. (b) Diversification reduces the portfolio's expected return because it reduces the portfolio's total risk. (c) As more securities are added to a portfolio, total risk would typi- cally be expected to fall at a decreasing rate. (d) The risk-reducing benefits of diversification do not occur mean- ingfully until at least 30 individual securities are included in the portfolio.Problem Set 2 1. (10 points) Annie and David are painting their apartment. At the paint store, David says he prefers Canary Yellow to Bumblebee Yellow, Lime Yellow, and Crayola Yellow. Annie finds new paint samples and asks David to compare Canary Yellow to School Bus Yellow and to Sunrise Yellow. David prefers Sunrise Yellow to Canary Yellow, and prefers School Bus Yellow to Canary Yellow. He also prefers Sunrise Yellow to School Bus Yellow. The store is out of Sunrise Yellow, so they buy School Bus Yellow and paint their apartment with it. David then insists that they go back, buy Lime Yellow, and repaint the apartment. True/False/Uncertain: David has rational preferences (as we define them). Problem 1 courtesy of William Wheaton. Used with permission. 2. (20 points) In each of the following examples, a consumer purchases just two goods: I and y. Based on the information in each of the following parts, sketch a plausible set of indifference curves (that is, draw at least two curves on a set of labeled axes, and indicate the direction of higher utility). Also, write down a utility function u(z, y) consistent with your graph. Note that although all these preferences should be assumed to be complete and transitive (as required for utility representation), not all will be monotone. (a) (4 points) Jessica enjoys bagels r and coffee y, and consuming more of one makes consuming the other more enjoyable. (b) (4 points) Plamen loves mocha swirl ice cream z, but he hates mushrooms y. (c) (4 points) Jennifer likes Cheerios r, and neither likes nor dislikes Frosted Flakes y. (d) (4 points) Edward always buys three white tank tops r for every pair of jeans y. (e) (4 points) Nancy likes both peanut butter I and jelly y, and always gets the same additional satisfaction from an ounce of peanut butter as she does from two ounces of jelly. 3. (20 points) A consumer's preferences are representable by the following utility function: u(x, y) = x3 + y. (a) (10 points) Obtain the MRS of the consumer at an arbitrary point (5. 7), where = > 0 and y > 0. (b) (10 points) Suppose the price of the second good (y) is 1, and the price of the first good (r) is denoted by p > 0. If the consumer's income is m > 0, obtain the optimal consumption bundle of the consumer (in terms of m and p). [Caution: make sure you cover cases in which m is relatively low, as well as cases in which m is relatively high.] 4. (25 points) It is exactly 24 hours before Lauren's physics final. She has an economics final directly after the physics final and has no time to study in between. Lauren wants to be a physicist, so she places more weight on her physics test score. Her utility function is given by u(p. e) = 0.6 1n(p) + 0.4In(e) where p is the score on the physics final and e is the score on the economics final. Although she cares more about physics, she is better at economics; for each hour spent studying economics she will increase her score by 3 points, but her physics score will only increase by 2 points for every hour spent studying physics. Studying zero hours results in a score of zero on both subjects (although In(0) is not defined, assume her utility for a score of zero is negative infinity). (a) (5 points) What constraints does Lauren face in her test score maximization problem? (b) (5 points) How many hours should Lauren optimally spend studying physics? How many hours studying economics? (hours are divisible) (c) (5 points) What economics and physics test scores will she achieve (ie. what are c' and p*)? (d) (5 points) What utility level will she achieve? (e) (5 points) Suppose Lauren can get an economics tutor. If she goes to the tutor, she will increase her economics test score by 5 points for every hour spent studying instead of 3 points, but will lose 4 hours of study time by going to the tutor. She cannot study while at the tutor, and going to the tutor does not directly improve her test score. Should Lauren go to the tutor? Problem 4 courtesy of William Wheaton. Used with permission