Ec 1745 Problem Set 4

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 our client would increase the portfolio’s returns and/or lower the variance of the portfolio. (See graph for point) c) E(Rp) = 1.2(.15)-.2(.05) = .17 → 17% σp = 1.2(.2) = .24 (See graph for point) d) w* = ((E(r1)-rF)/(kσ 2(r1))) → w* = ((.15-.05)/(10(.22))) = .25 E(Rp) = .25(.15)+.75(.05) = .075 → 7.5% σp = .25(.2) = .05 e) S = E(R-Rf)/σ Sb = (.1-.05)/.1 = .5 Sc = (.17-.05)/.24 = .5 Sd = (.075-.05)/.05 = .5

2. a) wm = (.05-.03)/(.1-.03) = .2857 βi = (.05-.03)/(.1-.03) = .2857 σ2(Rp) = (.2862)(.04) = .003265 b) Since both the efficient portfolio and asset i have the same expected return (E(r)), we can compare the differences in utility by looking at the differences in the variance. Investor 1: ΔE(U) = - .5(.09-.00327) = . 04337 Investor 2: ΔE(U) = (.09-.00327) = .08673 Thus, Investor 2, who has a risk aversion of k2 = 2 benefits more from this move. c) Investor 1: w* = ((.1-.03)/(.04)) = 1.75 E(Rp) = 1.75(.1)-.75(.03) = .1525 → 15.25% σ2p = (1.75)2(.04) = .1225 Investor 2: w* = ((.1-.03)/(2(.04))) = .875 E(Rp) = .875(.1)+.125(.03) = .09125 → 9.125% σ2p = (.875)2(.04) = .0306 d) Investor 1: ΔE(U) = (.1525-.5(.1225)) – (.05-.5(.003265)) = .04288 Investor 2: ΔE(U) = (.09125-.030625) - (.05-.003265) = -.01389 Investor 1 gains more by moving to his own preferred portfolio because the efficient portfolio in part a) is already closer to Investor 2’s preferred portfolio, thus moving to the preferred portfolio has less of an impact on utility for Investor 2 as it does for Investor 1. 3. a) 1) ρim = 0 → cov(ri,rm) = 0 → βi = 0 → E(ri) = rf+βi(E(rm)-rf) → E(ri) =rf = 3% 2) E(ri) = rf+βi(E(rm)-rf) = .03+1.2(.1-.03) = .114 → 11.4% 3) If future markets are efficient then the prices in the future market would be the same as those in the spot market, therefore we would see: E(ri) = rf+βi(E(rm)-rf) = .03+1.2(.1-.03) = .114 → 11.4% b) αi = E(ri)- rf-βi(E(rm)-rf) = .6(-1)+.4(($40000/$11000)-1)-.05-2(.15-.05) = .2045 → αi>0 → therefore you should take the project and set sail 4. a & b) (β’s on line 1, α’s on line 2) F ood 0.77 3 0.19 4

M ines 0.82 8 0.05 7

Oil 0.87 5 0.16 9

Clths 1.1 -0.089

D urbl 0.91 8 0.00 5

C hem s 1.02 3 0.07 9

Cnsu m 0.748 0.259

c) E(αi) = 0.026533 Median(αi) = 0.015544.

Cnstr 1.172 -0.07 6

Stee l 1.31 5 -0.2 07

FabPr

M achn

0.967

1.208

-0.029

0.004

C ars 1.20 4 0.08 2

Tran s 1.158 -0.11 9

Util s 0.81 4 0.02 6

Rtail

Finan

Other

0.955

1.139

0.888

0.072

0.016

0.008

The distribution of α’s is right skewed. There is a large outlier in consumption, and both E(αi) and Median(αi ) are greater than 0. Food AvgRet

1.017 Rtail 1.016

Mines 0.916 Finan 1.082

Oil

Clths

Durbl

Chems

Cnsum

Cnstr

FabPr

Machn

Cars

Trans

1.06

0.952

0.924

1.068

1.065

1.012

0.923

1.116

1.192

0.96

Other 0.907

Rm

Rf

0.974

0.308

Utils 0.876

d)

e)

The industrial portfolios do not conform to the “theoretical” SML; however, the prediction is still fairly accurate and would become more so if the risk-free rate were higher (which would lead to a higher intercept and, by decreasing Rm-Rf, a smaller slope).