Ec 1745 Problem Set 4

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Tarun Singh Worked with Cliff Beltzer Ec 1745 Problem Set 4 1. a) b) Yes, we can recommend a portfolio that is better than the one our client is currently holding. Our client is currently not on the efficient frontier, so we can move our client’s portfolio to this frontier by having the new portfolio lie between (.06, .08) and (.1, .1) along the efficient portfolio. This would correspond to the market portfolio having a weight of anywhere from 30% to 50%. By enacting the aforementioned changes
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Tarun SinghWorked with Cliff Beltzer Ec 1745 Problem Set 41.a)b) Yes, we can recommend a portfolio that is better than the one our client iscurrently holding. Our client is currently not on the efficient frontier, so wecan move our client’s portfolio to this frontier by having the new portfolio liebetween (.06, .08) and (.1, .1) along the efficient portfolio. This wouldcorrespond to the market portfolio having a weight of anywhere from 30% to50%. By enacting the aforementioned changes our client would increase theportfolio’s returns and/or lower the variance of the portfolio. (See graph forpoint) c) E(R p ) =   1.2(.15)-.2(.05) = .17 → 17%   σ p = 1.2(.2) = .24(See graph for point) d) w* = ((E(r 1 )-r F )/(kσ 2 (r 1 ))) → w* = ((.15-.05)/(10(.2 2 ))) = .25E(R p ) = .25(.15)+.75(.05) = .075 → 7.5%   σ p = .25(.2) = .05 e) S = E(R-R f  )/σ   S b = (.1-.05)/.1 = .5S c = (.17-.05)/.24 = .5S d = (.075-.05)/.05 = .5  2.a) w m = (.05-.03)/(.1-.03) = .2857   β i = (.05-.03)/(.1-.03) = .2857σ 2 (R p ) = (.286 2 )(.04) = .003265 b) Since both the efficient portfolio and asset i have the same expectedreturn (E(r)), we can compare the differences in utility by looking at thedifferences in the variance.   Investor 1: ΔE(U) = - .5(.09-.00327) = . 04337   Investor 2: ΔE(U) = (.09-.00327) = .08673Thus, Investor 2, who has a risk aversion of k 2 = 2 benefits more from thismove. c) Investor 1:   w* = ((.1-.03)/(.04)) = 1.75E(R p ) = 1.75(.1)-.75(.03) = .1525 → 15.25%   σ 2p = (1.75) 2 (.04) = .1225Investor 2:w* = ((.1-.03)/(2(.04))) = .875E(R p ) = .875(.1)+.125(.03) = .09125 → 9.125%   σ 2p = (.875) 2 (.04) = .0306 d) Investor 1: ΔE(U) = (.1525-.5(.1225)) – (.05-.5(.003265)) = .04288   Investor 2: ΔE(U) = (.09125-.030625) - (.05-.003265) = -.01389Investor 1 gains more by moving to his own preferred portfolio becausethe efficient portfolio in part a) is already closer to Investor 2’s preferredportfolio, thus moving to the preferred portfolio has less of an impact onutility for Investor 2 as it does for Investor 1. 3.a)   1) ρ im = 0 → cov(r i ,r m ) = 0 → β i = 0 → E(r i ) = r f  + β i (E(r m )-r f  ) → E(r i ) =r f  = 3%  2) E(r i ) = r f  + β i (E(r m )-r f  ) = .03+1.2(.1-.03) = .114 → 11.4% 3) If future markets are efficient then the prices in the future marketwould be the same as those in the spot market, therefore we would see:E(r i ) = r f  + β i (E(r m )-r f  ) = .03+1.2(.1-.03) = .114 → 11.4% b) α i = E(r i )- r f  - β i (E(r m )-r f  ) = .6(-1)+.4((\$40000/\$11000)-1)-.05-2(.15-.05) =.2045→ α i >0 → therefore you should take the project and set sail 4.a & b) ( β’s on line 1, α’s on line 2) FoodMinesOilClthsDurblChemsCnsumCnstrSteelFabPrMachnCarsTransUtils Rtail Finan Other  0.7730.8280.8751.10.9181.0230.7481.1721.3150.9671.2081.2041.1580.814 0.955 1.139 0.888 0.1940.0570.169-0.0890.0050.0790.259-0.076-0.207-0.0290.0040.082-0.1190.026 0.072 0.016 0.008 c) E(α i ) = 0.026533Median(α i ) = 0.015544.   The distribution of α’s is right skewed. There is a large outlier in consumption,and both E(α i ) and Median(α i ) are greater than 0. d)e)  The industrial portfolios do not conform to the “theoretical” SML; however, theprediction is still fairly accurate and would become more so if the risk-free rate werehigher (which would lead to a higher intercept and, by decreasing R m -R f  , a smallerslope). Food Mines Oil Clths Durbl Chems Cnsum Cnstr FabPr Machn Cars Trans Utils AvgRet 1.017 0.916 1.06 0.952 0.924 1.068 1.065 1.012 0.923 1.116 1.192 0.96 0.876 Rtail Finan Other Rm Rf  1.016 1.082 0.907 0.974 0.308
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