Problem Set II: Foundations of Finance Solutions 1 Foundations of Finance Prof. Alex Shapiro For problems 3,4,6,7, see solutions manual for BKM. For problems 1,2,5,8,9, see below: 1. a. Because the initial margin is 50%: 0.5 = NW/(Value of stock sold short) = 10,000/(Value of stock sold short) Value of stock sold short = 10,000 /0.5 = $20,000. So, can short at most 400 shares. b. Our balance sheet looks like this: Assets Liabilities Cash 30,000 20,000 (short stock, 400 shares, market value) 10,000 NW At the end of the year: Assets Liabilities Cash 30,000 21,200 (short stock, 400 shares, market value at 53) Accrued Interest 1,200 400 (dividends payable on the shorted stock) 9,600 NW The rate of return is –4%. 2. a. Er2=10.3%; σ2 = 10.8% b. The covariance between 1 and 2 is (1/3)(5-8.3)(-5-10.3)+(1/3)(5-8.3)(18-10.3)+(1/3)(15-8.3)(18-10.3)=25.56 ρ = 25.56/(4.7 × 10.8) = 0.5 c. Erp = (1/3)8.3 + (2/3)10.3 = 9.6% σp2 = (1/3)2(4.7)2 + (2/3)2(10.8)2 + 2(1/3)(2/3)25.56 = 65.65 σp = 8.1% d. y=0.6, so ErC = (0.6 × 9.6) +(0.4 × 5) = 7.76%, and σC = 0.6 × 8.1 = 4.86%Problem Set II: Foundations of Finance Solutions 25. Refer to the Lecture Notes on Uncertainty and the Lecture Notes on Optimal Risky Portfolios for all the moments. Suppose you form a portfolio of the Small Firm “asset,” Microsoft, and T-bills, with the T-bill, with rT-bill=0.323%. What are the preferred weights of the two risky assets in the risky portfolio? Graphical analysis (based on the efficient frontier in the notes) suggests that the risky asset portfolio you want to hold has positive weights invested in the small firm asset and in Microsoft (since [ is between + and x on the portfolio possibility curve for the small firm asset and Microsoft). Can calculate the exact weight of the small firm asset in the tangency portfolio using the moments given in the notes and the following formula (from the notes as well): ][E][][E][][E][][E][][E][][E][222Rr,r - Rr + Rr ,r - RrRr ,r - Rr = wSmallMsftSmallMsftSmallMsftMsftSmallSmallMsftMsftMsftSmallSmallMsftTSmall,σσσσσσ Now: E[RSmall] = E[rSmall] - rT-bill = 1.912 - 0.323 = 1.589. E[RMsft] = E[rMsft] - rT-bill = 3.126 - 0.323 = 2.803. σ2[rSmall] = 3.711 × 3.711 = 13.772. σ2 [rMsft] = 8.203 × 8.203 = 67.289. >rMsft, rSmall] = 12.030. 1.589 12.030 - 2.803 13.772 + 2.803 12.030 - 1.589 67.289 2.803 12.030 - 1.589 67.289 ×××××× = wTSmall, = 73.202 / [73.202 + 19.487] = 0.790. What are the preferred weights of the risky portfolio T and the riskless asset in the individual s portfolio? Depends on the tastes and preferences of the particular individual. Suppose you want to invest 75% in the tangency portfolio T (denoted by [ in the plots) and 25% in T-bills. What is the weight of the small firm asset and of Microsoft in your total portfolio? Use the following formula for the total (complete) portfolio C: wi,C = wi,T wT,C where wi,C is the weight of risky asset i in the total portfolio C. wi,T is the weight of risky asset i in the tangency portfolio T. wT,C is the weight of portfolio T in the total portfolio C (another notation we used for this weight is y). So, the answer is: wSmall,C = wSmall,T wT,C = 0.79 × 0.75 = 0.5925. wMsft,C = wMsft,T wT,C = 0.21 × 0.75 = 0.1575.Problem Set II: Foundations of Finance Solutions 3 8. a. βi = σiM/σM2 = 0.00045 / (0.03)2 = 0.50 b. E[ri] = rf + βi (E[rM] - rf) = 0.09 + 0.5 × 0.03 = 0.105 or 10.5%. c. Rearrange the SML to express βi as a function of E[ri]. βi = (E[ri] - rf) / (E[rM] - rf) = (0.10 - 0.09) / (0.12- 0.09) = 0.333 σiM = βiσM2 = 0.333 × 0.032 = 0.0003 d. βi = (0.15-0.09)/(0.12-0.09) = 0.06/0.03 = 2 σiM = 2 × 0.032 = 0.0018 9. a. Rearrange the SML to express E[rM] as E[rM] = rf + (E[ri] - rf) /βi . Therefore, E[rM] = rf + (E[rA] - rf) /βA = rf + (0.08 - rf)/0.4 E[rM] = rf + (E[rB] - rf) /βB = rf + (0.16 - rf)/1.2 Use the above two equations to solve for rf: rf + (0.08 - rf)/0.4 = rf + (0.16 - rf)/1.2, so rf = 0.04 or 4%. Hence, E[rM] = 0.04+ (0.08 - 0.04)/0.4 = 0.14 or 14%. b. βM = 1, by definition. c. σM2 = σAM / βA = 0.00225 / 0.4 = 0.005625, σM =0.075 or 7.5%. d. βC = σCM/σM2 = (ρCM σC σM)/σM2 = (ρCM σC) /σM = (ρAM σC) /σM = = ((σAM / (σA σM)) σC) /σM = (σAM σC) / (σA σM2)= = (0.00225 × 0.1125) / (0.06 × 0.005625) = 0.75 So, E[rC] = 0.04 + 0.75 (0.14 – 0.04) = 0.115 or 11.50% e. βB = σBM/σM2 = (ρBM σB σM)/σM2 = (ρBM σB ) / σM . Therefore, σB = (βB σM) / ρBM = (1.2 × 0.075) / 0.9 = 0.10 or 10%. f. Asset E[ri] σi βi A 8% 6% 0.40 C 11.5% 11.25% 0.75 M 14% 7.5% 1.00 B 16% 10% 1.20 We just confirmed that there is indeed a linear-positive relationship between E[ri] and βi , but there is no such relationship between E[ri] and σi .Problem Set II: Foundations of Finance Solutions 4Answers to Suggested Problems S1. If we buy the bill (at the ask), the settlement price is (1-0.058×150/360)=0.9758 (per $1 par). The BEY is (1-0.9758)/0.9758 × (365/150) = 6.03% (APR with n=150 day compounding). The EAR is FVIF(6.03%/M, M) - 1 (where M=365/150), so EAR=6.14%. If we sell the bill (at the bid), the settlement price is 0.9752. The BEY is 6.19%. The EAR is 6.30% (so we need the bank to offer more to justify selling). The EAR of the bank account is FVIF(5.95%/4, 4)-1 = 6.08%. Therefore, we should withdraw our money from the bank and buy a T-bill. (We give up an EAR of 6.08% to get an EAR of 6.14%.) S2. a. The situation is this: When the investor is entirely at stock 1, Er=10% and σ=10%. A portfolio composed of stock 2 and rf with the same risk can have a higher expected return. To construct this portfolio, let y be the amount in stock 2. Then 10%=σC=yσ2=y(15): y=2/3. The expected return is ErC=(1/3)5 + (2/3)20 = 15%. b. No, there are no guarantees. In any given year, stock 1 might do much better than stock 2, in which case the return on the new portfolio might be lower than the return on stock 1 alone. S3. a. b. In the portfolio of risky assets, P, wA=2/3 and wB=1.3, so σc = (3/4) 23.2 = 17.4% c. ErA=5+0.8(8)=11.4% d. The market model says rA-rf =1+0.8(rM-rf)+eA, or ErA-rf=1+0.8(ErM-rf), (since eA is expected to be zero). Since rf =5% and ErM-rf =8%, ErA-5=1+.8(8)=12.4% e. The difference might arise because the CAPM is completely wrong, because