Portfolio Theory 1. The expected returns and standard deviation of returns for two securities are as follows: Security Z Security Y Expected Return 15% 35% Standard Deviation 20% 40% The correlation between the returns is + .25. (a) Calculate the expected return and standard deviation for the following portfolios: i. all in Z ii..75 in Z and .25 in Y iii..5 in Z and .5 in Y iv..25 in Z and .75 in Y v. all in Y (b) Draw the investment opportunity set for these two risky assets. 1 2. Suppose that a fund that tracks the S&P has mean E(Rm)=16% and standard deviation σM=10%, and suppose that the T-bill rate Rf=8%. Answer the following questions about efficient portfolios: (a) What is the expected return and standard deviation of a portfolio that is totally invested in the risk-free asset? (b) What is the expected return and standard deviation of a portfolio that has 50% of its wealth in the risk-free asset and 50% in the S&P? (c) What is the expected return and standard deviation of a portfolio that has 125% of its wealth in the S&P, financed by borrowing 25% of its wealth at the risk-free rate? (d) What are the weights for investing in the risk-free asset and the S&P that produce a standard deviation for the entire portfolio that is twice the standard deviation of the S&P? What is the expected return on that portfolio? 3. Consider the following data: Expected Return Standard Deviation Russell Fund 16% 12% Windsor Fund 14% 10% S&P Fund 12% 8% The correlation between the returns on the Russell Fund and the S&P Fund is .7. The rate on Tbills is 6%. Which of the following portfolios would you prefer to hold in combination with Tbills and why? (a) Russell Fund (b) Windsor Fund (c) S&P Fund (d) A portfolio of 60% Russell Fund and 40% S&P Fund. 4. Excel Question. Use the portfolio optimizer posted on Sakai. Find out what happens to the share of asset 3 in the optimal risky portfolio (the tangency portfolio) in the following cases. Explain in words why you think this happens. (a) The expected return of asset 3 is increased. 2 (b) The standard deviation of asset 3 is increased. (c) The standard deviation of asset 2 is increased. (d) The correlation between assets 2 and 5 is increased from 0 to 50%. The Capital Asset Pricing Model 5. Assume the risk free rate equals Rf=4%, and the return on the market portfolio has expectation E[RM]=12% and standard deviation σM=15%. (a) What is the equilibrium risk premium (that is, the excess return on the market portfolio)? (b) If a certain stock has a realized return of 14%, what can we say about the beta of this stock? (c) If a certain stock has an expected return of 14%, what can we say about the beta of this stock? 6. You are given the following two equations: 𝐸(𝑅𝑖 ) = 𝑅𝑓 + (𝐸(𝑅𝑀 ) − 𝑅𝑓 )𝛽𝑖 𝐸(𝑅𝑃 ) = 𝑅𝑓 + ( 𝐸(𝑅𝑀 )−𝑅𝑓 𝜎𝑀 ) 𝜎𝑃 You also have the following information: E(RM)=.15, Rf=.06, σM=.15. Answer the following questions, assuming that the capital asset pricing model is correct: (a) Which equation would you use to determine the expected return on an individual security with a standard deviation of returns =.5 and a β=2? Given the parameters above, what is the expected return for that security? (b) Which equation would you use to determine the expected return on a portfolio knowing that it is an efficient portfolio (consisting of the market portfolio M combined with the risk-free rate)? If you were told that the standard deviation of returns on that portfolio is equal to σ M and you were given the above parameters, what is the expected return on that portfolio? (c) Can you determine the β of the portfolio in (b)? 3
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