I need help on 2b and 2c

ECON 163: Economics of Investments University of California, Merced Professor Jason Lee Problem Set #8 Due in class Tuesday, May 5th [100 Possible Points] You may work in small groups [2 to 3 students] to complete the homework. If you chose to work in groups be sure to include the names of all the students on the homework. Only one assignment needs to be submitted per group. Question #1: Preferences and Portfolio Theory [12 Points] Consider an investor with preferences given by the utility function U = E(r) - 0.5A2 and there are two portfolios with the following characteristics: Portfolio A Portfolio B E(r) = 0.148 = 0.16 E(r) = 0.082 = 0.068 (a) Suppose that the investor has a level of risk aversion of A = 4. Which portfolio should the investor choose? [3 Points] (b) Suppose that the investor has a level of risk aversion of A = 6. Which portfolio should the investor choose? [3 Points] (c) Suppose the investor has a level of risk aversion of A = 4 What must the return be on a risk-free asset in order for the investor to be indifferent between investing in the risk-free asset and Portfolio B? [3 Points] (d) Suppose the investor has a level of risk aversion of A = 4. Calculate the risk-premium associated with Portfolio B. Assume that the risk-free rate is the same as your answer in Part(c). [3 Points] Question #2: Optimal Risky Portfolio [22 Points] You are trying to decide whether to buy Vanguard's Large Stock Equity Fund and/or its Treasury Bond Fund (both are risky assets). You believe that next year involves several possible scenarios to which you have assigned probabilities. You have also estimated the expected returns for each of the two funds for each scenario. Your spreadsheet looks like the following: Next Year's Possibilities Probability Large Stock Equity Fund Expected Rate of Return Bond Fund Expected Rate of Return Severe Recession 0.20 -0.30 -0.09 Mild Recession 0.30 -0.15 0.10 Normal Growth 0.45 0.12 0.06 Strong Growth 0.05 0.45 0.02 (a) Given the probabilities for the four possible economic conditions and their associated rates of return, calculate the expected return for each fund. [4 Points] (b) Given the probabilities for the four possible economic conditions and their associated rates of return, calculate the standard deviation for each fund. [6 Points] (c) Using your answers from Parts (a) and (b) calculate the covariance between the two funds, and the correlation coefficient. [6 Points] (d) Using the formulas for the expected return and risk of a portfolio, calculate the expected return and standard deviation of a portfolio that is weighted 40% in the stock equity fund and 60% in the bond fund. Show your work. [6 Points] Question #3 Graphing the Investment Opportunity Set [15 Points] Suppose that an investor has a choice between investing in a bond fund (B) and a stock fund (S). The bond fund has an expected return of E(rB) = 0.06 while the stock fund has an expected return of E(rS) = 0.10. The standard deviation of the bond fund is B= 0.12 and the standard deviation of the stock fund is S = 0.25. Assume the correlation coefficient is BS = 0.2. (a) Calculate the expected return and standard deviation for each of the following portfolio weights. If you are comfortable using EXCEL you can use the \"Portfolio Weight Calculator\" (located in Cat Courses) to complete the table. When inputting the values do not use decimals (e.g. The expected return of the bond fund is inputted as 6 not 0.06). You do not have to show your calculations. [4 Points] WS = Portfolio Weight in Equity Fund WB = Portfolio Weight in Bond Fund 0 1 0.1 0.9 0.2 0.8 0.3 0.7 0.4 0.6 0.5 0.5 0.6 0.4 0.7 0.3 0.8 0.2 0.9 0.1 1 0 Expected Return of Overall Portfolio Standard Deviation of Overall Portfolio (b) Plot the expected return and standard deviation of the various portfolios using EXCEL (draw the investment opportunity set). Identify the minimum variance portfolio (MV Portfolio) [6 Points] (c) Prove that an investor would never choose a portfolio that has a weight of 10% equity fund and 90% bond fund. [5 Points] Question #4: The Optimal Portfolio [20 Points] You are attempting to construct an optimal portfolio consisting of T-bills and a risky portfolio. The expected return on the risky portfolio is 15% and the standard deviation is 18.1%. The T-bill rate is 2.5%. (a) If you put 25% of your funds in T-bills and 75% in the risky portfolio, what is the expected return and standard deviation of your overall portfolio? [3 Points] (b) Construct a Capital Allocation Line for a portfolio consisting of the T-bills and the risky portfolio. Draw the line and indicate the y-intercept and the point where the portfolio is entirely made up of risky assets. What is the slope of this line? [3 Points] (c) Suppose that you have a degree of risk aversion of A = 5.6. What is the optimal portfolio for this investor? In other words, what is the optimal weight (y) in the risky portfolio and what is the optimal weight (1-y) in T-bills? [5 Points] (d) What is the expected return and standard deviation on your optimal portfolio? Illustrate this point on your CAL graph you drew in Part (b). [4 Points] (e) Now suppose Maria has a degree of risk aversion of A = 4.0. What is the optimal weight (y) in the risky portfolio and what is the optimal weight (1-y) in T-bills for Maria? Briefly explain why Maria's optimal weight in the risky portfolio differs from yours? [5 Points] Question #5: Choosing the Optimal Risky Portfolio [12 Points] A pension fund manager is considering three assets. The first is a risky stock with high growth potential (Stock A), the second is a dividend-yielding lower risk stock with more modest growth potential (Stock B), and the third are safe short-term government bonds that yields a guaranteed rate of 3.0%. The probability distribution of the risky stocks are Fund Expected Return Standard Deviation Stock A 8% 10% Stock B 15% 18% Assume that the correlation between the fund returns is 0.30. Calculate the optimal risky portfolio. What fraction of funds in the optimal risky portfolio should be allocated to Stock A? What fraction of funds in the optimal risky portfolio should be allocated to Stock B? Question #6: CAPM [19 Points] Use the Figure below to answer Parts (a) - (c) E(r) SML 0.0882 0.075 0.031 0 1 1.3 Beta () (a) What is the expected return of the market portfolio? No explanation is required. [2 Points] (b) What is the risk-free rate (rf)? No explanation is required. [2 Points] (c) What is the expected return of an asset with a beta of 1.3? [5 Points] Use the following scenario to answer Part (d) Suppose that the current price for a share of Six Flags (SIX) stock is at $35.96. The stock is expected to pay $3.32 in dividends next year and you anticipate that you will be able to sell the stock for $37 next year. Assume that E(rM) = 0.10, the risk-free rate (rf) = 0.042 and SIX has a beta () = 0.93. (d) Is SIX stock overpriced or underpriced at its current price. You must justify your answer. [Hint: Find the required rate of return of SIX stock and then compare it to the 1 Year HPR of SIX stock] [10 Points]