NYU COR1-GB 2311 - Foundations of Finance problem set 3

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Solution Set 3 Foundations of Finance1Problem Set 3 Solution.I. Expected Return, Return Standard Deviation, Covariance and Portfolios (cont):State Probability Asset A Asset B Riskless AssetBoom 0.25 24% 14% 7%Normal Growth 0.5 18% 9% 7%Recession 0.25 2% 5% 7%A. What is the expected return and standard deviation of return of a portfolioconsisting of ω% invested in asset A and (1-ω)% in the riskless asset when ω% is1. -20%?2. 60%?3. 120%?As an illustration, for ωA,p = -0.2:E[Rp]= ωA,p E[RA] + (1- ωA,p ) Rf = -0.2 x 15.5% + 1.2 x 7% = 5.3%andσ[Rp]= |ωA,p | σ[RA]= |-0.2| 8.1701% = 1.6340%ωA,pE[Rp] σ[Rp]-0.2 5.3% 1.6340%0.6 12.1% 4.9020%1.2 17.2% 9.8041%B. What is the expected return and standard deviation of return of a portfolioconsisting of ω% invested in asset B and (1-ω)% in the riskless asset when ω% is1. -20%?2. 60%?3. 120%?As an illustration, for ωB,p = 1.2:E[Rp]= ωB,p E[RB] + (1- ωB,p ) Rf = 1.2 x 9.25% + -0.2 x 7% = 9.7%andσ[Rp]= |ωB,p | σ[RB]Solution Set 3 Foundations of Finance2= |1.2| 3.1918% = 3.8301%ωB,pE[Rp] σ[Rp]-0.2 6.55% 0.6383%0.6 8.35% 1.9151%1.2 9.7% 3.8301%C. If a risk-averse investor has to decide whether to hold either asset A with theriskless asset or asset B with the riskless asset, which asset would the investorprefer to hold in combination with the riskless asset? Explain why? Do you needmore information about the investor’s preferences to answer the question? Any risk averse individual prefers the risky asset whose CAL has the higher slope. The reason isthat for any point on the lower sloped CAL, there exists a point on the higher sloped CAL withthe same expected return but lower standard deviation.slope-CAL(A) = {E[RA]-Rf}/σ[RA] = {15.5%-7%}/8.1701% = 1.0404.slope-CAL(B) = {E[RB]-Rf}/σ[RB] = {9.25%-7%}/3.1918% = 0.7049.So any risk averse individual prefers to hold asset A in combination with the riskless asset thanasset B. D. What is the expected return and standard deviation of return of a portfolioconsisting of ω% invested in asset A and (1-ω)% in asset B when ω% is1. -20%?2. 80%?3. 120%?As an illustration, for ωA,p = 0.8:E[Rp]= ωA,p E[RA] + (1- ωA,p ) E[RB] = 0.8 x 15.5% + 0.2 x 9.25% = 14.25%andσ[Rp]2= ωA,p 2 σ[RA]2 + ωB,p 2 σ[RB]2 + 2 ωA,p ωB,p σ[RA, RB]= (0.8x0.8) 66.75 + (0.2x0.2) 10.1875 + 2 (0.8x0.2) 24.125 = 42.72 + 0.4075 + 7.72 = 50.8475σ[Rp] = 7.1307%.Solution Set 3 Foundations of Finance3ωA,pE[Rp] σ[Rp]-0.2 8% 2.4%0.8 14.25% 7.1307%1.2 16.75% 9.2167%Solution Set 3 Foundations of Finance4II. Using Dividend Yield Information (cont): Suppose the following data is to be used by MsQ (a risk-averse investor) to form a portfolio that consists of the small firm fund and T-bills.E[RSmall(t)] = 1.369% σ[RSmall(t)] = 8.779%E[DP(start t)] = 4.446% σ[DP(start t)] = 1.513%σ[DP(start t),RSmall(t)] = 1.967where DP(start t) is the dividend yield on the S&P 500 known at the start of month t. RSmall(t) is the return on the small firm fund in month t.A. Suppose it is the end of March 1997, Ms Q does not know DP and the return onT-bills for April is 0.3%.1. Will Ms Q short sell the small firm fund? The expected April return on the small firm fund is:E[RSmall(t)] = 1.369%. Ms Q wants to lie on the positive-sloped portion of the portfolio possibility curve. E[RSmall(t)]>Rf. So Ms Q does not want to short sell.2. Will Ms Q buy the small firm fund on margin?Ms Q wants to lie on the positive-sloped portion of the portfolio possibility curve. E[RSmall(t)]>Rf. So Ms Q may want to buy on margin depending on how risk averse she is.3. Will Ms Q buy a positive amount of both assets? Ms Q wants to lie on the positive-sloped portion of the portfolio possibility curve. E[RSmall(t)]>Rf. So Ms Q may want to buy positive amounts of both depending on how riskaverse she is.B. Suppose it is the end of March 1997, Ms Q knows that DP is 2% and the return onT-bills for April is 0.3%.1. Will Ms Q short sell the small firm fund? Given Ms Q’s information, the expected April return on the small firm fund is:µSmall,DP + φSmall,DP DP(start Apr) = -2.451 + 0.859 x 2 = -0.733%.Ms Q wants to lie on the positive-sloped portion of the portfolio possibility curve. E[RSmall(t)]<Rf. So Ms Q does want to short sell.Solution Set 3 Foundations of Finance52. Will Ms Q buy the small firm fund on margin?Ms Q wants to lie on the positive-sloped portion of the portfolio possibility curve. E[RSmall(t)]<Rf. So Ms Q does not want to buy on margin.3. Will Ms Q buy a positive amount of both assets? Ms Q wants to lie on the positive-sloped portion of the portfolio possibility curve. E[RSmall(t)]<Rf. So Ms Q does not want to buy positive amounts of both.C. Suppose it is the end of October 1997, Ms Q does not know DP and the return onT-bills for November is 0.4%.1. Will Ms Q short sell the small firm fund? 2. Will Ms Q buy the small firm fund on margin?3. Will Ms Q buy a positive amount of both assets? The answer to this question is the same as for part A.D. Suppose it is the end of October 1997, Ms Q knows that DP is 5% and the returnon T-bills for November is 0.4%.1. What is the expected November return on the small firm fund?µSmall,DP + φSmall,DP DP(start Nov) = -2.451 + 0.859 x 5 = 1.844%.2. Will Ms Q short sell the small firm fund? Given Ms Q’s information, the expected November return on the small firm fund is:µSmall,DP + φSmall,DP DP(start Nov) = -2.451 + 0.859 x 5 = 1.844%.Ms Q wants to lie on the positive-sloped portion of the portfolio possibility curve. E[RSmall(t)]>Rf. So Ms Q does not want to short sell.3. Will Ms Q buy the small firm fund on margin?Ms Q wants to lie on the positive-sloped portion of the portfolio possibility curve. E[RSmall(t)]>Rf. So Ms Q may want to buy on margin depending on how risk averse she is.4. Will Ms Q buy a positive amount of both assets? Ms Q wants to lie on the positive-sloped portion of the portfolio possibility curve. E[RSmall(t)]>Rf. So Ms Q may want to buy positive amounts of both depending on how riskaverse she is.Solution Set 3 Foundations of Finance6III. The Two Risky Asset Case:A. As an illustration, for ωS,p = 0.6:E[Rp]= ωS,p E[RS] + (1- ωS,p ) E[RB] = 0.6 x 22% + 0.4 x 13% = 18.4%andσ[Rp]2= ωS,p 2 σ[RS]2 + ωB,p 2 σ[RB]2 + 2 ωS,p ωB,p σ[RS, RB]= ωS,p 2 σ[RS]2 + ωB,p 2 σ[RB]2 + 2 ωS,p ωB,p ρ[RS, RB] σ[RS] σ[RB]= (0.6x0.6) (32x32) + (0.4x0.4) (23x23) + 2 (0.6x0.4) (0.15x32x23) = 368.64 + 84.64 + 52.992 = 506.272σ[Rp] =


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NYU COR1-GB 2311 - Foundations of Finance problem set 3

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