Financial Management, Answers to Assignments 2

6-2 T-bill rate = r* + IP 5.5% = r* + 3.25% r* = 2.25%. 6-3 r* = 3%; I1 = 2%; I2 = 4%; I3 = 4%; MRP = 0; rT2 = ?; rT3 = ? r = r* + IP + DRP + LP + MRP. Since these are Treasury securities, DRP = LP = 0. rT2 = r* + IP2. IP2 = (2% + 4%)/2 = 3%. rT2 = 3% + 3% = 6%. rT3 = r* + IP3. IP3 = (2% + 4% + 4%)/3 = 3.33%. rT3 = 3% + 3.33% = 6.33%. 6-5 r* = 3%; IP2 = 3%; rT2 = 6.2%; MRP2 = ? rT2 = r* + IP2 + MRP2 = 6.2% rT2 = 3% + 3% + MRP2 = 6.2% MRP2 = 0.2%. 6-7 rT1 = 5%; 1rT1 = 6%; rT2 = ? (1 + rT2)2 = (1.05)(1.06) (1 + rT2)2 = 1.113 1 + rT2 = 1.055 rT2 = 5.5%. 6-9 r7 = r* + IP7 + MRP7 + DRP + LP. r* = 0.03. IP7 = [0.03 + 0.04 + (5)(0.035)]/7 = 0.035. MRP7 = 0.0005(t – 1) = 0.0005(6) = 0.003. DRP = 0. LP = 0. rT7 = 0.03 + 0.035 + 0.003 = 0.068 = 6.8%. 6-10 Basic relevant equations: rt = r* + IPt + DRPt + MRPt + IPt. But here IPt is the only premium, so rt = r* + IPt. IPt = Avg. inflation = (I1 + I2 + . . .)/N. We know that I1 = IP1 = 3% and r* = 2%. Therefore, rT1 = 2% + 3% = 5%. rT3 = rT1 + 2% = 5% + 2% = 7%. But, rT3 = r* + IP3 = 2% + IP3 = 7%, so IP3 = 7% – 2% = 5%. We also know that It = Constant after t = 1. We can set up this table: 1 2 3 r* 2% 2% 2% I 3% I I IPt 3%/1 = 3% (3% + I)/2 = IP2 (3% + I + I)/3 = IP3 r = r* + IPt 5% r3 = 7%, so IP3 = 7% – 2% = 5%. IP3 = (3% + 2I)/3 = 5% 2I = 12% I = 6%. 6-11 We’re given all the components to determine the yield on the bonds except the default risk premium (DRP) and MRP. Calculate the MRP as 0.1%(5 – 1) = 0.4%. Now, we can solve for the DRP as follows: 7.75% = 2.3% + 2.5% + 0.4% + 1.0% + DRP, or DRP = 1.55%. 6-12 First, calculate the inflation premiums for the next three and five years, respectively. They are IP3 = (2.5% + 3.2% + 3.6%)/3 = 3.1% and IP5 = (2.5% + 3.2% + 3.6% + 3.6% + 3.6%)/5 = 3.3%. The real risk-free rate is given as 2.75%. Since the default and liquidity premiums are zero on Treasury bonds, we can now solve for the maturity risk premium. Thus, 6.25% = 2.75% + 3.1% + MRP3, or MRP3 = 0.4%. Similarly, 6.8% = 2.75% + 3.3% + MRP5, or MRP5 = 0.75%. Thus, MRP5 – MRP3 = 0.75% – 0.40% = 0.35%. 7-5 a. 1. 5%: Bond L: Bond S: 2. 8%: Bond L: Bond S: 3. 12%: Bond L: Bond S: Input N = 15, I/YR = 5, PMT = 100, FV = 1000, PV = ?, PV = $1,518.98. Change N = 1, PV = ? PV = $1,047.62. From Bond S inputs, change N = 15 and I/YR = 8, PV = ?, PV = $1,171.19. Change N = 1, PV = ? PV = $1,018.52. From Bond S inputs, change N = 15 and I/YR = 12, PV = ?, PV = $863.78. Change N = 1, PV = ? PV = $982.14. b. Think about a bond that matures in one month. Its present value is influenced primarily by the maturity value, which will be received in only one month. Even if interest rates double, the price of the bond will still be close to $1,000. A 1-year bond’s value would fluctuate more than the one-month bond’s value because of the difference in the timing of receipts. However, its value would still be fairly close to $1,000 even if interest rates doubled. A long-term bond paying semiannual coupons, on the other hand, will be dominated by distant receipts, receipts that are multiplied by 1/(1 + rd/2)t, and if rd increases, these multipliers will decrease significantly. Another way to view this problem is from an opportunity point of view. A 1-month bond can be reinvested at the new rate very quickly, and hence the opportunity to invest at this new rate is not lost; however, the long-term bond locks in subnormal returns for a long period of time. 7-8 The rate of return is approximately 15.03%, found with a calculator using the following inputs: N = 6; PV = -1000; PMT = 140; FV = 1090; I/YR = ? Solve for I/YR = 15.03%. Despite a 15% return on the bonds, investors are not likely to be happy that they were called. Because if the bonds have been called, this indicates that interest rates have fallen sufficiently that the YTC is less than the YTM. (Since they were originally sold at par, the YTM at issuance= 14%.) Rates are sufficiently low to justify the call. Now investors must reinvest their funds in a much lower interest rate environment. 7-9 a. VB = N  (1INT r ) t 1 t d  M (1  rd ) N M = $1,000. PMT = 0.09($1,000) = $90. 1. VB = $829: Input N = 4, PV = -829, PMT = 90, FV = 1000, YTM = I/YR = ? I/YR = 14.99%. 2. VB = $1,104: Change PV = -1104, YTM = I/YR = ? I/YR = 6.00%. b. Yes. At a price of $829, the yield to maturity, 15%, is greater than your required rate of return of 12%. If your required rate of return were 12%, you should be willing to buy the bond at any price below $908.88. 7-10 a. Solving for YTM: N = 9, PV = -901.40, PMT = 80, FV = 1000 I/YR = YTM = 9.6911%. b. The current yield is defined as the annual coupon payment divided by the current price. CY = $80/$901.40 = 8.875%. Expected capital gains yield can be found as the difference between YTM and the current yield. CGY = YTM – CY = 9.691% – 8.875% = 0.816%. Alternatively, you can solve for the capital gains yield by first finding the expected price next year. N = 8, I/YR = 9.6911, PMT = 80, FV = 1000 PV = -$908.76. VB = $908.76. Hence, the capital gains yield is the percentage price appreciation over the next year. CGY = (P1 – P0)/P0 = ($908.76 – $901.40)/$901.40 = 0.816%. c. As rates change they will cause the end-of-year price to change and thus the realized capital gains yield to change. As a result, the realized return to investors will differ from the YTM. 7-13 The problem asks you to solve for the YTM and Price, given the following facts: N = 5  2 = 10, PMT = 80/2 = 40, and FV = 1000. In order to solve for I/YR we need PV. However, you are also given that the current yield is equal to 8.21%. Given this information, we can find PV (Price). Current yield = Annual interest/Current price 0.0821 = $80/PV PV = $80/0.0821 = $974.42. Now, solve for the YTM with a financial calculator: N = 10, PV = -974.42, PMT = 40, and FV = 1000. Solve for I/YR = YTM = 4.32%. However, this is a periodic rate so the nominal YTM = 4.32%(2) = 8.64%. 7-15 First, we must find the amount of money we can expect to sell this bond for in 5 years. This is found using the fact that in five years, there will be 15 years remaining until the bond matures and that the expected YTM for this bond at that time will be 8.5%. N = 15, I/YR = 8.5, PMT = 90, FV = 1000 PV = -$1,041.52. VB = $1,041.52. This is the value of the bond in 5 years. Therefore, we can solve for the maximum price we would be willing to pay for this bond today, subject to our required rate of return of 10%. N = 5, I/YR = 10, PMT = 90, FV = 1041.52 PV = -$987.87. VB = $987.87. You would be willing to pay up to $987.87 for this bond today. 8-1 r̂ = (0.1)(-50%) + (0.2)(-5%) + (0.4)(16%) + (0.2)(25%) + (0.1)(60%) = 11.40%. 2 = (-50% – 11.40%)2(0.1) + (-5% – 11.40%)2(0.2) + (16% – 11.40%)2(0.4) + (25% – 11.40%)2(0.2) + (60% – 11.40%)2(0.1) 2  = 712.44;  = 26.69%. CV = 26.69% = 2.34. 11.40% 8-2 Investment $35,000 40,000 Total $75,000 Beta 0.8 1.4 bp = ($35,000/$75,000)(0.8) + ($40,000/$75,000)(1.4) = 1.12. 8-3 rRF = 6%; rM = 13%; b = 0.7; r = ? r = rRF + (rM – rRF)b = 6% + (13% – 6%)0.7 = 10.9%. 8-7 Portfolio beta = $400,000 $600,000 $1,000,000 (1.50) + (-0.50) + (1.25) + $4,000,000 $4,000,000 $4,000,000 $2,000,000 (0.75) $4,000,000 bp = (0.1)(1.5) + (0.15)(-0.50) + (0.25)(1.25) + (0.5)(0.75) = 0.15 – 0.075 + 0.3125 + 0.375 = 0.7625. rp = rRF + (rM – rRF)(bp) = 6% + (14% – 6%)(0.7625) = 12.1%. Alternative solution: First, calculate the return for each stock using the CAPM equation [rRF + (rM – rRF)b], and then calculate the weighted average of these returns. rRF = 6% and (rM – rRF) = 8%. Stock A B C D Total Investment $ 400,000 600,000 1,000,000 2,000,000 $4,000,000 Beta 1.50 (0.50) 1.25 0.75 r = rRF + (rM – rRF)b 18% 2 16 12 rp = 18%(0.10) + 2%(0.15) + 16%(0.25) + 12%(0.50) = 12.1%. 8-8 In equilibrium: rJ = r̂J = 12.5%. rJ = rRF + (rM – rRF)b 12.5% = 4.5% + (10.5% – 4.5%)b b = 1.33. Weight 0.10 0.15 0.25 0.50 1.00 8-9 We know that bR = 1.50, bS = 0.75, rM = 13%, rRF = 7%. ri = rRF + (rM – rRF)bi = 7% + (13% – 7%)bi. rR = 7% + 6%(1.50) = 16.0% rS = 7% + 6%(0.75) = 11.5 4.5% 8-13 a. Using Stock X (or any stock): 9% = rRF + (rM – rRF)bX 9% = 5.5% + (rM – rRF)0.8 (rM – rRF) = 4.375%. b. bQ = 1/3(0.8) + 1/3(1.2) + 1/3(1.6) bQ = 0.2667 + 0.4000 + 0.5333 bQ = 1.2. c. rQ = 5.5% + 4.375%(1.2) rQ = 10.75%. d. Since the returns on the 3 stocks included in Portfolio Q are not perfectly positively correlated, one would expect the standard deviation of the portfolio to be less than 15%. D (1  g) $5 [1  (0.05)] D1 $5 (0.95) $4.75  0     $23.75. rs  g rs  g 0.15  (0.05) 0.15  0.05 0.20 9-10 P̂0  9-13 The problem asks you to determine the value of P̂3 , given the following facts: D1 = $2, b = 0.9, rRF = 5.6%, RPM = 6%, and P0 = $25. Proceed as follows: Step 1: Calculate the required rate of return: rs = rRF + (rM – rRF)b = 5.6% + (6%)0.9 = 11%. Step 2: Use the constant growth rate formula to calculate g: r̂s  D1 g P0 $2 g $25 g  0.03  3%. 0.11  Step 3: Calculate P̂3 : P̂3 = P0(1 + g)3 = $25(1.03)3 = $27.3182  $27.32. Alternatively, you could calculate D4 and then use the constant growth rate formula to solve for P̂3 : D4 = D1(1 + g)3 = $2.00(1.03)3 = $2.1855. P̂3 = $2.1855/(0.11 – 0.03) = $27.3182  $27.32. 9-14 Calculate the dividend cash flows and place them on a time line. Also, calculate the stock price at the end of the supernormal growth period, and include it, along with the dividend to be paid at t = 5, as CF5. Then, enter the cash flows as shown on the time line into the cash flow register, enter the required rate of return as I/YR = 15, and then find the value of the stock using the NPV calculation. Be sure to enter CF0 = 0, or else your answer will be incorrect. D0 = 0; D1 = 0; D2 = 0; D3 = 1.00; D4 = 1.00(1.5) = 1.5; D5 = 1.00(1.5)2 = 2.25; D6 = 1.00(1.5)2(1.08) = $2.43. P̂0 = ? 0 | rs = 15% 1 | 2 | 3 4 5 6 | | | | gs = 50% gs = 50% gn = 8% 1.00 1.50 2.25 2.43  1/(1.15)3  1/(1.15)4  1/(1.15)5 0.658 0.858 18.378 +34.714 = 2.43 0.15  0.08 36.964 $19.894 = P̂0 P̂5 = D6/(rs – g) = $2.43/(0.15 – 0.08) = $34.714. This is the stock price at the end of Year 5. CF0 = 0; CF1-2 = 0; CF3 = 1.0; CF4 = 1.5; CF5 = 36.964; I/YR = 15. With these cash flows in the CFLO register, press NPV to calculate the value of the stock today: NPV = $19.89. 10-6 a. rs = D1 $2.14 +g= + 7% = 9.3% + 7% = 16.3%. $23 P0 b. rs = rRF + (rM – rRF)b = 9% + (13% – 9%)1.6 = 9% + (4%)1.6 = 9% + 6.4% = 15.4%. c. rs = Bond rate + Risk premium = 12% + 4% = 16%. d. Since you have equal confidence in the inputs used for the three approaches, an average of the three methodologies probably would be warranted. rs = 10-7 a. rs = 16.3%  15.4%  16% = 15.9%. 3 D1 +g P0 $3.18 + 0.06 $36 = 14.83%. = b. F = ($36.00 – $32.40)/$36.00 = $3.60/$36.00 = 10%. c. re = D1/[P0(1 – F)] + g = $3.18/$32.40 + 6% = 9.81% + 6% = 15.81%. 10-8 Debt = 40%, Common equity = 60%. P0 = $22.50, D0 = $2.00, D1 = $2.00(1.07) = $2.14, g = 7%. rs = D1 $2.14 +g= + 7% = 16.51%. P0 $22.50 WACC = (0.4)(0.12)(1 – 0.4) + (0.6)(0.1651) = 0.0288 + 0.0991 = 12.79%. 10-10 If the investment requires $5.9 million, that means that it requires $3.54 million (60%) of common equity and $2.36 million (40%) of debt. In this scenario, the firm would exhaust its $2 million of retained earnings and be forced to raise new stock at a cost of 15%. Needing $2.36 million in debt, the firm could get by raising debt at only 10%. Therefore, its weighted average cost of capital is: WACC = 0.4(10%)(1 – 0.4) + 0.6(15%) = 11.4%.