424midtermSolutionsFall2008

424midtermSolutionsFall2008 - University of Washington...

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University of Washington Fall 2008 Department of Economics Eric Zivot Economics 424 Midterm Exam This is a closed book and closed note exam. However, you are allowed one page of notes (double-sided). Answer all questions and write all answers in a blue book or on separate sheets of paper. Time limit is 1 hours and 50 minutes. Total points = 100. I. Return Calculations (20 pts, 5 points each) 1. Consider a one year investment in two assets: Amazon stock and the S&P 500 index. Suppose you buy Amazon and S&P 500 at the end of September 2007 at 75 . 1526 , 15 . 93 1 , 1 , = = t S t A P P and then sell at the end of September 2008 for 74 . 1164 , 76 . 72 , , = = t S t A P P . (Note: these are actual closing prices taken from Yahoo!) a. What are the simple annual returns for the two stocks? > pa.1 = 93.15 > pa.2 = 72.76 > ps.1 = 1526.75 > ps.2 = 1164.74 # a) simple returns on Amazon and sp500 > ra = (pa.2 - pa.1)/pa.1 > rs = (ps.2 - ps.1)/ps.1 > ra [1] -0.2188943 > rs [1] -0.2371115 b. What are the continuously compounded annual returns for the two stocks? > log(1 + ra) [1] -0.2470447 > log(1 + rs) [1] -0.2706434 c. The annual inflation rate between September 2007 and September 2008 was about 5%. Using this information, determine the simple and continuously compounded real annual returns on Amazon and S&P 500. > inflat = 0.05 > # simple real returns

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> ra.real = (1+ra)/(1+inflat) - 1 > rs.real = (1+rs)/(1+inflat) - 1 > ra.real [1] -0.2560898 > rs.real [1] -0.2734395 > # cc real returns > log(1+ra.real) [1] -0.2958349 > log(1+rs.real) [1] -0.3194336 d. At the end of September, 2006, suppose you have \$100,000 to invest in Amazon and S&P 500 over the next year. Suppose you sell short \$60,000 in S&P 500 and use the proceeds to buy \$160,000 in Amazon. Using the results from part a, compute the annual simple and continuously compounded return on the portfolio. > xs = -60000/100000 > xa = 160000/100000 > xa [1] 1.6 > xs [1] -0.6 > rp = xa*ra + xs*rs > rp [1] -0.2079639 > # cc portfolio return > log(1 + rp) [1] -0.2331483 II. Probability Theory and Matrix Algebra (20 points, 5 points each) 1. Suppose you currently hold \$2M (million) in Starbucks stock. That is, your initial wealth at the beginning of the month is 0 \$2 WM = . Let R sbux denote the monthly simple return on Starbucks stock, and assume that 2 ~ (0.03,(0.20) ) sbux RN . Let 10 (1 ) SBUX WW R =+ be a random variable representing your wealth at the end of the month. a) Compute 1 [] EW , 11 var( ) and ( ) WS D W > w0 = 2 > e.rsbux = 0.03 > sd.rsbux = 0.20 > e.w = w0 + w0*e.rsbux > e.w [1] 2.06
> var.w = w0*(w0*sd.rsbux*sd.rsbux) > var.w [1] 0.16 > sd.w = w0*sd.rsbux > sd.w [1] 0.4 b) What is the probability distribution of 1 W ? Sketch the distribution, indicating the location of 1 [] EW and 11 []2 () E W SD W ± . Since R is normally distributed and W 1 is a linear function of R, W 1 is also normally distributed with mean \$2.06M and SD \$0.4M. > e.w + 2*sd.w [1] 2.86 > e.w - 2*sd.w [1] 1.26 1.0 1.5 2.0 2.5 3.0 0.0 0.2 0.4 0.6 0.8 1.0 Normal distribution for Wealth w.vals pdf

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c) Briefly explain why the normal distribution may not be appropriate for describing the distribution of simple returns. The normal distribution is defined from to ∞∞ . Simple returns cannot be smaller than -1. Also, multi-period simple returns are multiplicative (geometric average). That is, the 2 period return is a geometric average (multiplicative) of two 1 period returns. If the 1 period returns are normally distributed then the 2 period return will not be normal.
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