What is the cost of the restriction in terms of Sharpe’s measure? What is the utility loss to the investor (A = 1.8) given his new complete portfolio?Calculate the following for a portfolio manager who is not allowed to short sell securities. If allowed to short sell securities, the manager's Sharpe ratio is 0.3647. What is the utility loss to the investor (A = 1.8) given his new complete portfolio? Note: Do not round intermediate calculations. Round your answers to 2 decimal places. Calculate using numbers in decimal form, not percentages. For example use "20" for calculation if standard deviation is provided as 20%. Show less CasesUtility LevelsUnconstrained%Constrained%Passive%
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To calculate the utility loss to the investor given the constraints on short selling, we will follow these steps: Show more…
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You are considering an investment in the stock market. In the stock market, there are two risky stocks (A and B) and a risk-free claim, C (you can think of it as the T-bill). The covariances and returns of these three stocks are described in the following table: Covariance A B C A 0.25 0.06 0 B 0.06 0.15 0 C 0 0 0 Return 12.50% 9.75% 4.00% Assume that you have a mean-variance utility function with risk aversion A=5. That is, your utility function is U. Let P be the risky portfolio that consists of stocks A and B. Let wPA be the weight of stock A in portfolio P and let wPB = 1 - wPA be the weight of stock B in the portfolio P. Write down the Sharpe ratio of portfolio P as a function of wPA. Compute the weight of the optimal risky portfolio, wP*A, that maximizes the Sharpe ratio. (Hint: See equation 7.13) However, instead of maximizing the Sharpe ratio directly, you decide to use the Markowitz Portfolio Optimization Model. In the next couple of questions, we will compute the optimal risky portfolio using the Markowitz procedure. As before, let wPA be the weight of stock A in portfolio P and let wPB = 1 - wPA be the weight of stock B in the portfolio P. a. First, we want to find the weights of the optimal risky portfolio that minimizes the variance. i. Write down the Lagrangian of this minimization problem as a function of wPA and wPB. ii. Write down the first-order conditions. iii. Write down the system of equations in matrix form. iv. Solve the system. In this question, you are expected to solve the system in matrix form by hand (I urge you to do it!). Please include (only) the solution. (Use Excel to double-check your answer). Hint: In this case, the matrix A^-1 is: 1.785714 -1.78571 0.321429 A^-1 = -1.78571 1.785714 0.678571 0.321429 0.678571 -0.24214 v. Is the solution similar to the solution in b)? Explain. b. Second, we want to compute the weights of the optimal risky portfolio that minimizes the variance subject to a portfolio return of 10.50%. Write down the Lagrangian of this minimization problem as a function of wPA and wPB. Write down the first-order conditions. Write down the system of equations in matrix form. iv. Solve the system using Excel. v. Is the solution similar to the solution in b)? Explain. c. Third, we want to compute the weights of the optimal portfolio on the efficient frontier that has the highest Sharpe ratio. i. What is the "standard" name of this portfolio? ii. Compute the weights of this portfolio. iii. Is the solution similar to the solution in b)? Explain. d) Using the solution in c), c, iii), compute the weights of your optimal portfolio. (i.e., compute the weight of T-bills and the risky portfolio in your optimal portfolio).
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The following is part of the computer output from a regression of monthly returns on Waterworks stock against the S&P 500 index. A hedge fund manager believes that Waterworks is underpriced, with an alpha of 2.4% over the coming month. Standard Deviation: 0.95 Beta: 0.65 R-square of Residuals: 0.1 (i.e., 10% monthly) Now suppose that the manager misestimates the beta of Waterworks stock, believing it to be 0.5 instead of 0.95. The standard deviation of the monthly market rate of return is 9%. If he holds a $2,000,000 portfolio of Waterworks stock, the S&P 500 currently is at 2,000 and the contract multiplier is $50. a. What is the standard deviation of the (now improperly) hedged portfolio? (Round your answer to 3 decimal places.) b. What is the probability of incurring a loss on the improperly hedged portfolio over the next month if the monthly market return has an expected value of 1% and a standard deviation of 9%? The manager holds a $2 million portfolio of Waterworks stock and wishes to hedge market exposure for the next month using 1-month maturity S&P 500 futures contracts. The S&P 500 currently is at 2,000 and the contract multiplier is $50. Assume the risk-free rate is 0.5% per month. (Round your answer to 2 decimal places. Enter your answer as percentages and not as numbers. (Eg: Enter "12%" and not "0.12").)
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