424midtermFall2009solutions-1

424midtermFall2009solutions-1 - University of Washington...

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University of Washington Fall 2009 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 = 84. I. Return Calculations (16 pts, 4 points each) Consider a one year investment in two assets: the Vanguard S&P 500 index (VFINX) and the Vanguard Short Term Bond mutual fund (VBISX). Suppose you buy one share of the S&P 500 fund and one share of the bond fund at the end of September, 2008 for 500, 1 , 1 104.61, 9.70 sp t bond t PP −− == , and then sell these shares at the end of September, 2009 for 500, , 97.45, 10.46 sp t bond t . (Note: these are actual closing prices taken from Yahoo!) a. What are the simple annual returns for the two investments? > p.sp500.1 = 104.61 > p.sp500.2 = 97.45 > p.bond.1 = 9.70 > p.bond.2 = 10.46 # a) simple returns on sp500 and bond > r.sp500 = (p.sp500.2 - p.sp500.1)/p.sp500.1 > r.bond = (p.bond.2 - p.bond.1)/p.bond.1 > r.sp500 [1] -0.06844 > r.bond [1] 0.07835 b. What are the continuously compounded annual returns for the two investments? # b) cc returns on Amazon and sp500 > log(1 + r.sp500) [1] -0.0709 > log(1 + r.bond) [1] 0.07543
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c. Assume you get the same annual returns from part a. every year for the next 10 years. How much will $10,000 invested in each fund be worth after 10 years? > w0 = 10000 > w1.sp500 = w0*(1 + r.sp500)^10 > w1.bond = w0*(1 + r.bond)^10 > w1.sp500 [1] 4921 > w1.bond [1] 21262 d. The annual inflation rate between September 2008 and September 2009 was about -1% (yes, we actually had deflation !). Using this information, determine the simple and continuously compounded real annual returns on S&P 500 and the bond fund. > inflat = -0.01 # simple real returns > r.sp500.real = (1+r.sp500)/(1+inflat) - 1 > r.bond.real = (1+r.bond)/(1+inflat) - 1 > r.sp500.real [1] -0.05904 > r.bond.real [1] 0.08924 # cc real returns > log(1+r.sp500.real) [1] -0.06085 > log(1+r.bond.real) [1] 0.08548 II. Probability Theory and Matrix Algebra (16 points, 4 points each) Let r t denote the monthly continuously compounded return on an asset and suppose that 2 ~ (0.01,(0.05) ) t ri i d N . a. What is the relationship between the continuously compounded return r t and the simple return R t ? Given this relationship, what is the probability distribution of 1 + R t ? ln(1 ) 1 1 tt rr t t rR R e R e =+ = + = Because 2 ~ (0.01,(0.05) ) t i d N , it follows that 2 1 ~ lognormal(0.01,(0.05) ) t R + .
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b. Give an expression for the 6-month continuously compounded return, r t (6), in terms of the monthly continuously compounded returns. Using this expression compute [ (6)], var( (6)), and ( (6)). tt t Er r SDr What is the probability distribution of r t (6)? 5 15 0 55 00 2 (6) [ (6)] [ ] 0.01 6 (0.01) 0.06 var( (6)) var var( ) 6 (0.05) 0.015 ( (6)) 6(0.05) 0.015 0.1224 t t t j j j jj j t j t rr r r r r SD r −− = == =+ + + = = × = ⎛⎞ = × = ⎜⎟ ⎝⎠ = ∑∑ " Therefore , 2 (6) ~ (0.06,(0.1224) ) t rN Let R i denote the continuously compounded return on asset i ( i = 1,2,3) with E [ R i ] = μ i , var( R i ) = 2 i σ and cov( R i , R j ) = ij . Define the 3 × 1 vectors = = = = = 1 1 1 , , , , 3 2 1 3 2 1 3 2 1 3 2 1 1 y x μ R y y y x x x R R R
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424midtermFall2009solutions-1 - University of Washington...

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