Ec 1745 Problem Set 3

Tarun Singh Worked with Cliff Beltzer Ec 1745 Problem Set 3 1. a) $900k = (.55($900k(1+r))+(.45($500k)))/(1.10) → r = .5454 = 54.54% b) $1000k = (.55($1000k(1+r))+(.45($500k)))/(1.10) → r = .5909 = 59.09% 2. a) PV(B) = (((.5($50M))+(.5($20M))) /(1) = $35M PV(E) = (((.5($50M))+(.5($0)))/(1) = $25M b) DP = (($50M/$35M)-1) = .4286 = 42.86% c) Return (Bond in Sunshine) = (($50M/$35M)-1) = .4286 = 42.86% Return (Bond in Rain) = (($20M/$35M)-1) = -.4286 = -42.86% d) Return (Equity in Sunshine) = (($50M/$25M)-1) = 1 = 100% Return (Equity in Rain) = (($0/$25M)-1) = -1 = -100% e) i) Bond: Sunshine- $50M , Rain- $30M Equity: Sunshine- $40M, Rain- $0 ii) PV(B) = (((.5($50M))+(.5($30M)))/(1) = $40M PV(E) = (((.5($40M))+(.5($0)))/(1) = $20M iii) Return (Bond in Sunshine) = (($50M/$40M)-1) = .25 = 25% Return (Bond in Rain) = (($30M/$40M)-1) = -.25 = -25% Return (Equity in Sunshine) = (($40M/$20M)-1) = 1 = 100% Return (Equity in Rain) = (($0/$20M)-1) = -1 = -100% iv) The owners would not be tempted to reorganize because reorganization would only lower the expected positive returns on bonds and would have no impact on the returns to equity. f) i) Bond: Sunshine- $50M , Rain- $10M Equity: Sunshine- $60M, Rain- $0 ii) PV(B) = (((.5($50M))+(.5($10M)))/(1) = $30M PV(E) = (((.5($60M))+(.5($0)))/(1) = $30M iii) Return (Bond in Sunshine) = (($50M/$30M)-1) = .6667 = 66.67% Return (Bond in Rain) = (($10M/$30M)-1) = -.6667 = -66.67% Return (Equity in Sunshine) = (($60M/$30M)-1) = 1 = 100% Return (Equity in Rain) = (($0/$30M)-1) = -1 = -100% iv) The owners would be tempted to reorganize because reorganization would only increase the expected positive returns on bonds and would have no impact on the expected returns to equity. 3. a) E(RBonds) = ((.3(.16))+(.4(.06))+(.3(-.04))) = .06 = 6% Var(RBonds) = ((.3(.16-.06)2)+(.4(.06-.06)2)+(.3(-.04-.06)2)) = .006 S.D.(RBonds) = √(.006) = .07746 = 7.746% E(RStocks) = ((.3(-.11))+(.4(.13))+(.3(.27))) = .10 = 10% Var(RStocks) = ((.3(-.11-.10)2)+(.4(.13-.10)2)+(.3(.27-.10)2)) = .02226 S.D.(RStocks) = √(.02226) = .149198 = 14.9198% b) Cov(RB,Rs) = ((.3(-.21)(.10))+(.4(.03)(0))+(.3(.17)(-.10))) = -.0114

Cov(RB,Rs) = ρ(S.D.(RStocks)) (S.D.(RBonds)) Corr(RB,Rs) = ρ = -.0114/((.07746)(.149198)) = -.986426 c) E(RPort) = ((.5(.06))+(.5((.10))) = .08 = 8% Var(RPort) = ((.52)(.006))+ ((.52)(.02226))+ (2(.5)(.5)(-.0114))) = .001365 S.D.(RPort) = √(.001365) = .036946 = 3.6946% d) An investor with these preferences prefers the equal-weighted portfolio to bonds because no matter what the value of k is, the portfolio yields a higher return and has a lower variance than the bonds. Thus in order, to maximize his/her utility the investor ought to choose the equal-weighted portfolio assuming the risk aversion constant is positive. 4. a)

Looking at the graph I would guess that the portfolio where w=.55 has the lowest variance. b) Var(Rp) = w2Var(Ri)+(1-w)2Var(Rj)+2w(1-w)Cov(Ri,Rj) ∂Var(Rp)/∂w = 2w(Var(Ri))+2(1-w)(Var(Rj))+2(1-2w)Cov(Ri, Rj) 0 = 2w(Var(Ri))+2(1-w)(Var(Rj))+2(1-2w)Cov(Ri, Rj) →w = (Var(Rj)-Cov(Ri, Rj))/(Var(Ri)+Var(Rj)-2Cov(Ri,Rj)) Cov(Ri,Rj) = (-.9)(.010.5) (.015.5) = -.011 → w = (.015-(-.011))/(.010+.015-2(-.011)) = .55314 = 55.314%