S183 M1 S10 - Using the package stockPortfolio we have...

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University of California, Los Angeles Department of Statistics Statistics C183/C283 Instructor: Nicolas Christou Exam 1 07 May 2010 Name: Problem 1 (20 points) The betas for 10 stocks in two historical periods 2000-2004 and 2005-2009 are as follows: beta1 beta2 [1,] 0.9072828 0.7333601 [2,] 1.0874136 1.0096048 [3,] 0.9871119 1.1143148 [4,] 1.0084073 1.1011334 [5,] 0.7606293 0.7711888 [6,] 0.8047901 0.7834646 [7,] 0.9533157 0.9914738 [8,] 0.8036708 1.1083840 [9,] 1.0867607 0.9524100 [10,] 1.0315184 0.8759303 a. Explain how you can obtain an estimate for the beta of stock 8 for the period 2010-2014 using the Blume’s technique. b. Suppose that for the second period 2005-2009 the variance of the return of the S & P500 index is σ 2 m = 0 . 00217. Assume that the single index model holds. Find the covariance between stocks 1 and 3 during the same period. c. Explain how you can obtain an estimate for the beta of stock 8 for the period 2010-2014 using the Vacicek’s technique. d. Suppose that the correlation coefficient between stock A and S & P500 during the period 2005-2009 is 0.20. The variance of the return of stock A during the same period is 0.0143 and the variance of the return of the S & P500 index was σ 2 m = 0 . 00217. Find the beta of stock A .
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Problem 2 (20 points) Use the following for questions (a) and (b) below: Stock ¯ R σ A 0.12 0.20 B ??? 0.08 It is also given that ρ AB = 0 . 1. a. What expected return on stock B would result in an optimum portfolio of 1 2 A and 1 2 B ? Assume short sales are allowed and that R f = 0 . 04. b. What expected return on stock B would mean that stock B would not be held? Assume short sales are allowed and that R f = 0 . 04. c. Suppose X and Y represent the returns of two stocks. Show that these two random variables X and Y cannot possibly have the following properties: E ( X ) = 0 . 3 , E ( Y ) = 0 . 2 , E ( X 2 ) = 0 . 1 , E ( Y 2 ) = 0 . 29, and E ( XY ) = 0. Reminder: σ XY = E ( X - μ X )( Y - μ Y ) = EXY - ( EX )( EY ).
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Problem 3 (20 points) Using the package stockPortfolio we have obtained the returns of IBM, Exxon-Mobil, and the S&P500 index for the period 2005-01-31 to 2009-12-31 . The summary statistics, variance-covariance matrix of the
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