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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 (p,, denotes the correlation be- tween asset i and asset j, and M denotes the market portfolio): PIBM.M = 0.60 Paw,y = 0.80 UTBM = 0.0900 Jay - 0.0625 You also have the following data about the market portfolio M and the risk less asset F: FM - 0.13 FF - 0.04 03 - 0.04 Suppose that individuals can borrow and lead at ry and that the Capi- tal Asset Pricing Model (CAPM) describes expected returns on assets. You have $200, 000 invested in IBM. $200,000 invested in GM, and $100,000 invested in the riskless asset. (a) What are the expected rates of return on IBM stock and GM stock? (b) Assume that the correlation between IBM and GM, Pinarain, is 0.40. What is the variance of your portfolio? What is its beta, (c) Suppose that you can also invest in the market portfolio. Find an efficient portfolio that has the same standard deviation as your portfolio, but has the highest expected rate of return possible. What is the expected rate of return on this portfolio? 24 2005, Andere W. To anil Jiang Wang 1.6 Risk & Portfolio Choice 1 QUESTIONS 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 0 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 A B C D 11% 25% 1.0 .15 14.5% 35% .3 1.0 .45 .2 9% 30% 15 45 10 25 11.5% 32% .2 25 1.08. Which of the following portfolios can not be on the Markowitz efficient frontier? Explain briefly. Portfolio | Expected Return |Standard Deviation Q 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 TB TA .04 .025 .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 200s, 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: Stock | ER 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:E4. You want to hedge the DEM value of a CAD Im inflow using futures contracts. On Germany's exchange, there is a futures contract for USD 100,000 at DEM/USD 1.5. (a) Your assistant runs a bunch of regressions: (1) AS[DEM/CAD] = 01 + BI A/[USD/DEM]. (2) AS[DEM/CAD] = 02 + $2 4/[DEM/USD].2.2 ADDITIONAL PROBLEMS Additional problem 2.1 Consider a market model with two states and two assets. where: ! B(0) = 55. B(T.W1) =60. B(T.w,) = 60. S(0) = 45. S(T.w|) = 45, S(T.w,) = 40. . Is there an arbitrage opportunity? . Is there an arbitrage opportunity if shortselling is forbidden? . Is there an arbitrage opportunity if shortselling is allowed, but there is a 51.5 Options 1. A stock price is currently $50. It is known that at the end of two months it will be either $53 or $48. The risk-free interest rate is 10% per annum with continuous compounding. What is the value of a two- month European call option with a strike price of $49? 2. A stock price is currently $80. It is known that at the end of four months it will be either $75 or $85. The risk-free interest rate is 5% per annum with continuous compounding. What is the value of a four- month European put option with a strike price of $80? 3. Today's price of three traded call options on BackBay.com, all expiring in one month, are as follows: Strike Price Option Price $50 60 $3 70 $14 You are considering buying a "butterfly spread" consisting of the fol- lowing positions: . Buy 1 call at strike price of $50 . Sell (write) 2 calls at strike price of $60 . Buy 1 call at strike price of $70. (a) Plot the payoff of your total position for different values of the stock price on the maturity date. (b) What is the dollar investment required to establish the spread? (c) For what stock prices on the maturity date will you be making an overall profit? 4. You are given the following prices: Security Maturity (years) Strike Price ($) JEK stock 94 Put on JEK stock 50 3 Put on JEK stock 60 5 Call on JEK 50 Call on JEK 60 ? Tbill (FV=100) 91 What is the price of the two call options? D 208, Andrew W. Le and Jiang Wang 1.5 Options 1 QUESTIONS 5. (a) Here are the payoff diagrams of some popular trading strategies using just put and call options with same maturities. How would you replicate them? Identifyfigthe number and strikes of call or put options that have to be bought or sold in order to generate these payoffs. (All angles are 45 degrees!) Payoff by Stock Price Payoff Stock Price (c)