4. This problem examines expected utility maximization in the context of the mean-variance model. Suppose an...

  

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4. This problem examines expected utility maximization in the context of the mean-variance model. Suppose an investor's utility for a portfolio having expected return and standard deviation is given by u(,) = -o, where a is the investor's coefficient of absolute risk aversion. a. What is the investor's marginal rate of substitution between risk and return . Compute your answer assuming the graph has return on the vertical axis and risk on the horizontal. That is, MRS = b. Assume the risk free asset has a 4% return, and the market portfolio has an expected return of 12% and a standard deviation of 16%. Assuming the investor can divide their wealth between the risk free asset and the market portfolio, what is the equation of the Capital Markets Line? c. Suppose the investor chooses risk and return to maximize expected utility from along the Capital Markets Line. Sketch the optimal solution. d. Suppose that the investor's coefficient of absolute risk aversion is a = 1/16. What is the mean and standard deviation of the investor's optimal portfolio? What is the investor's marginal rate of substitution when they choose this portfolio? What fraction of the investor's wealth is invested in the risky asset? e. Suppose that the investor's coefficient of absolute risk aversion is a = 1/48. What is the mean and standard deviation of the investor's optimal portfolio? What is the investor's marginal rate of substitution when they choose this portfolio? What fraction of the investor's wealth is invested in the risky asset? f. How do the optimized marginal rates of substitution you found in parts d and e relate to the slope of the Capital Markets Line? How is it related to the claim that the slope of the Capital Markets Line is the market price of risk? e. How do your answers to parts d and e relate to the separation principle? 4. This problem examines expected utility maximization in the context of the mean-variance model. Suppose an investor's utility for a portfolio having expected return and standard deviation is given by u(,) = -o, where a is the investor's coefficient of absolute risk aversion. a. What is the investor's marginal rate of substitution between risk and return . Compute your answer assuming the graph has return on the vertical axis and risk on the horizontal. That is, MRS = b. Assume the risk free asset has a 4% return, and the market portfolio has an expected return of 12% and a standard deviation of 16%. Assuming the investor can divide their wealth between the risk free asset and the market portfolio, what is the equation of the Capital Markets Line? c. Suppose the investor chooses risk and return to maximize expected utility from along the Capital Markets Line. Sketch the optimal solution. d. Suppose that the investor's coefficient of absolute risk aversion is a = 1/16. What is the mean and standard deviation of the investor's optimal portfolio? What is the investor's marginal rate of substitution when they choose this portfolio? What fraction of the investor's wealth is invested in the risky asset? e. Suppose that the investor's coefficient of absolute risk aversion is a = 1/48. What is the mean and standard deviation of the investor's optimal portfolio? What is the investor's marginal rate of substitution when they choose this portfolio? What fraction of the investor's wealth is invested in the risky asset? f. How do the optimized marginal rates of substitution you found in parts d and e relate to the slope of the Capital Markets Line? How is it related to the claim that the slope of the Capital Markets Line is the market price of risk? e. How do your answers to parts d and e relate to the separation principle?