42 0106462 0106462 8312010 5133 0023399 0039124

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Unformatted text preview: following table contains Shell Oil Company’s monthly closing price, monthly returns, and monthly returns without dividends over July 2010 – December 2010: Shell Oil Company Date Price Return Return without Dividend 7/30/2010 53.42 0.106462 0.106462 8/31/2010 51.33 -0.023399 -0.039124 9/30/2010 58.79 0.145334 0.145334 10/29/2010 64.32 0.094064 0.094064 11/30/2010 60.31 -0.049285 -0.062345 12/31/2010 66.67 0.105455 0.105455 What was the dividend in August 2010, November 2010, and December 2010? State all assumptions and show all calculations. Answer The return between periods The term and are: is “capital gains” or return without dividends and the term is dividend yield. Now: Now: Shell Oil Company Date Price Return Return without Dividend Dividend Yield 7/30/2010 53.42 0.106462 0.106462 0 8/31/2010 51.33 -0.023399 -0.039124 0.015725 9/30/2010 58.79 0.145334 0.145334 0 10/29/2010 64.32 0.094064 0.094064 0 11/30/2010 60.31 -0.049285 -0.062345 0.01306 12/31/2010 66.67 0.105455 0.105455 0 Now: { } { } Thus: 42 ECO 204 Chapter 7: Practice Problems & Solutions for Economics of Financial Portfolio Allocation in ECO 204 (this version 2012-2013) University of Toronto, Department of Economics (STG). ECO 204, S. Ajaz Hussain. Do not distribute. Shell Oil Company Date Price Return Return without Dividend Dividend Yield Dividend 7/30/2010 53.42 0.106462 0.106462 0 8/31/2010 51.33 -0.023399 -0.039124 0.015725 0.84003 9/30/2010 58.79 0.145334 0.145334 0 0 10/29/2010 64.32 0.094064 0.094064 0 0 11/30/2010 60.31 -0.049285 -0.062345 0.01306 0.840019 12/31/2010 66.67 0.105455 0.105455 0 0 Thus the dividends in August 2010, November 2010 and December 2010 were: August 2010 = 0.84003 November 2010 = 0.840019 December 2010 = 0 (7.3) The following table contains the return on US 30 day T-Bills purchased on Dec 31st 2011 and the average and variance of Shell Oil Company’s monthly returns (August 2005 to December 2011): 30 Day US T-Bill return (issued Dec 31st 2011) 0.0002 Shell Oil Average Monthly return 0.0101 Shell Oil Variance of Monthly return 0.0056 (a) Consider an investor with a mean-variance utility function. What is the parameter of “risk intolerance” if she borrows 10% of her funds to invest in a portfolio of 30-day US T-Bills and Shell Oil stocks in December 2011? State all assumptions and show all calculations. Answer let’s assume the investor has the following utility function: Where c is the parameter of “risk intolerance”. Then the portfolio return and variance will be defined as: ( Taking expectation of ) and substituting it in to the utility function gives ( ) Therefore, we optimize the utility function with respect to 43 ECO 204 Chapter 7: Practice Problems & Solutions for Economics of Financial Portfolio Allocation in ECO 204 (this version 2012-2013) University of Toronto, Department of Economics (STG). ECO 204, S. Ajaz Hussain. Do not distribute. ( Notice that ) s.t. , so we can ignore the constraint in this particular case. Hence FOC gives Given Shell Oil Company return and variance, and substituting 1.1 for , c solves (b) The following table contains the average, variance, and covariance of Shell Oil Company’s and Research in Motion’s (RIM) monthly returns (August 2005 to December 2011) Shell Oil Average Monthly return 0.0101 Shell Oil Variance of Monthly return 0.0056 RIM Average Monthly return 0.0109 RIM Variance of Monthly return 0.0312 Covariance Shell Oil Company and RIM Monthly return 0.0043 Suppose an investor allocates funds between Shell Oil and RIM stocks. What fraction of this portfolio consists of RIM stocks if her goal is to minimize portfolio risk? State all assumptions and show all calculations. Answer Here we are trying to construct a portfolio that has minimum variance therefore we have to minimize the portfolio variance with respect to f. ( ( ) ) ( ) Hence ( ) ( ) FOC: ( Substituting the values from the table ) ( ) solves 44 ECO 204 Chapter 7: Practice Problems & Solutions for Economics of Financial Portfolio Allocation in ECO 204 (this version 2012-2013) University of Toronto, Department of Economics (STG). ECO 204, S. Ajaz Hussain. Do not distribute. (7.4) The following table contains the sample mean returns and sample standard deviations for MSFT and IBM returns, the sample covariance between MSFT and IBM returns, and the return on 30 day US T-Bills: Mar 31st 1986 - Dec 31st 2011 MSFT IBM Mean r 0.0247 0.0100 Risk σ 0.1071 0.0805 Cov(Dell, Boeing) 0.0038 rf, Dec 31st, 2011 0.0002 (a) Assume that an investor has the mean-variance utility function over expected portfolio returns and portfolio risk: ) in the risk free asset and a fraction in a risky portfolio (portfolio of 2 risky Suppose that they invest fraction ( assets). When allocating funds in the risky portfolio, they invest fraction ( ) in IBM and fraction in MSFT. Calculate the optimal proportion of funds they allocate in the risk free asset, in BM and in MSFT. Assume Answer: Please see Excel Model Chapter 7.2 for numerical computations. Between risk-free asset and risky portfolio: Solve the following UMP – ( ) [( ( ) ) ( ) Substituting the constraints in the original UMP and differentiating with respect to UMP: yields the following solution to the For the two-risky assets: Maximize the market price of risk – 45 ECO 204 Chapter 7: Practice Problems & Solutions for Economics of Financial Portfolio...
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