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FIN303 Answers to recommended problems Chapter 9 9-3 Yes. If a company decides to increase its payout ratio, then the dividend yield component will rise, but the expected long-term capital gains yield will decline. 9-4 Yes. The value of a share of stock is the PV of its expected future dividends. If the two investors expect the same future dividend stream, and they agree on the stock’s riskiness, then they should reach similar conclusions as to the stock’s value. 9-1 D0 = $1.50; g1-3 = 7%; gn = 5%; D1 through D5 = ? D1 = D0(1 + g1) = $1.50(1.07) = $1.6050. D2 = D0(1 + g1)(1 + g2) = $1.50(1.07)2 = $1.7174. D3 = D0(1 + g1)(1 + g2)(1 + g3) = $1.50(1.07)3 = $1.8376. D4 = D0(1 + g1)(1 + g2)(1 + g3)(1 + gn) = $1.50(1.07)3(1.05) = $1.9294. D5 = D0(1 + g1)(1 + g2)(1 + g3)(1 + gn)2 = $1.50(1.07)3(1.05)2 = $2.0259. 9-2 D1 = $0.50; g = 7%; rs = 15%; P̂0 = ? P̂0 9-3 9-4 D1 $0.50 $6.25. rs g 0.15 0.07 P0 = $20; D0 = $1.00; g = 6%; P̂1 = ?; rs = ? P̂1 = P0(1 + g) = $20(1.06) = $21.20. r̂s = D1 $1.00(1.06) +g= + 0.06 P0 $20 = $1.06 + 0.06 = 11.30%. rs = 11.30%. $20 a. The terminal, or horizon, date is the date when the growth rate becomes constant. This occurs at the end of Year 2. b. 0 | 1.25 rs = 10% gs = 20% 1 | 1.50 gs = 20% 2 | gn = 5% 1.80 3 | 1.89 1 37.80 = 1.89 0.10 0.05 The horizon, or terminal, value is the value at the horizon date of all dividends expected thereafter. In this problem it is calculated as follows: $1.80 (1.05) $37.80. 0.10 0.05 c. The firm’s intrinsic value is calculated as the sum of the present value of all dividends during the supernormal growth period plus the present value of the terminal value. Using your financial calculator, enter the following inputs: CF0 = 0, CF1 = 1.50, CF2 = 1.80 + 37.80 = 39.60, I/YR = 10, and then solve for NPV = $34.09. 9-5 The firm’s free cash flow is expected to grow at a constant rate, hence we can apply a constant growth formula to determine the total value of the firm. Firm value = FCF1/(WACC – g) = $150,000,000/(0.10 – 0.05) = $3,000,000,000. To find the value of an equity claim upon the company (share of stock), we must subtract out the market value of debt and preferred stock. This firm happens to be entirely equity funded, and this step is unnecessary. Hence, to find the value of a share of stock, we divide equity value (or in this case, firm value) by the number of shares outstanding. Equity value per share = Equity value/Shares outstanding = $3,000,000,000/50,000,000 = $60. 9-11 Each share of common stock is worth $60, according to the corporate valuation model. First, solve for the current price. P̂0 = D1/(rs – g) = $0.50/(0.12 – 0.07) = $10.00. If the stock is in a constant growth state, the constant dividend growth rate is also the capital gains yield for the stock and the stock price growth rate. Hence, to find the price of the stock four years from today: P̂4 = P0(1 + g)4 2 9-16 = $10.00(1.07)4 = $13.10796 ≈ $13.11. 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 | 1 | rs = 15% 0.658 0.858 18.378 $19.894 = P̂0 2 | 3 4 | g = 50% | s 1.00 1.50 5 6 | g = 8% | n 2.25 2.43 2.43 1/(1.15)3 +34.714 = 0.15 0.08 1/(1.15) 4 1/(1.15)5 36.964 = D6/(rs – g) = $2.43/(0.15 – 0.08) = $34.714. This is the stock price at the end of Year 5. P̂5 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 get the value of the stock today: NPV = $19.89. 9-17 a. Terminal value = b. 0 | WACC = 13% ($ 17.70) 23.49 522.10 $527.89 1/1.13 $42.80 $40 (1.07) = = $713.33 million. 0.13 0.07 0.06 1 | -20 1/(1.13)2 1/(1.13)3 2 | 30 3 | 40 Vop 3 gn = 7% 4 | 42.80 = 713.33 753.33 Using a financial calculator, enter the following inputs: CF0 = 0; CF1 = -20; CF2 = 30; CF3 = 753.33; I/YR = 13; and then solve for NPV = $527.89 million. 3 c. Total valuet=0 = $527.89 million. Value of common equity = $527.89 – $100 = $427.89 million. Price per share = $427 .89 = $42.79. 10.00 Chapter 10 10-1 rd(1 – T) = 0.12(0.65) = 7.80%. 10-3 40% Debt; 60% Common equity; rd = 9%; T = 40%; WACC = 9.96%; rs = ? WACC = (wd)(rd)(1 – T) + (wc)(rs) 0.0996 = (0.4)(0.09)(1 – 0.4) + (0.6)rs 0.0996 = 0.0216 + 0.6rs 0.078 = 0.6rs rs = 13%. 10-4 P0 = $30; D1 = $3.00; g = 5%; rs = ? a. rs = D1 $3.00 +g= + 0.05 = 15%. P0 $30.00 b. F = 10%; re = ? re = D1 $3.00 +g = + 0.05 P0 (1 F) $30(1 0.10) = 10-8 $3.00 + 0.05 = 16.11%. $27.00 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%. 4 Chapter 11 11-1 Financial calculator solution: Input CF0 = -52125, CF1-8 = 12000, I/YR = 12, and then solve for NPV = $7,486.68. 11-2 Financial calculator solution: Input CF0 = -52125, CF1-8 = 12000, and then solve for IRR = 16%. 11-4 Since the cash flows are a constant $12,000, calculate the payback period as: $52,125/$12,000 = 4.3438, so the payback is about 4 years. 11-5 Project K’s discounted payback period is calculated as follows: Annual Discounted @12% Period Cash Flows Cash Flows Cumulative 0 ($52,125) ($52,125.00) ($52,125.00) 1 12,000 10,714.29 (41,410.71) 2 12,000 9,566.33 (31,844.38) 3 12,000 8,541.36 (23,303.02) 4 12,000 7,626.22 (15,676.80) 5 12,000 6,809.12 (8,867.68) 6 12,000 6,079.57 (2,788.11) 7 12,000 5,428.19 2,640.08 8 12,000 4,846.60 7,486.68 The discounted payback period is 6 + 11-6 $2,788. 11 years, or 6.51 years. $5,42 8.19 a. Project A: Using a financial calculator, enter the following: CF0 = -25, CF1 = 5, CF2 = 10, CF3 = 17, I/YR = 5; NPV = $3.52. Change I/YR = 5 to I/YR = 10; NPV = $0.58. Change I/YR = 10 to I/YR = 15; NPV = -$1.91. Project B: Using a financial calculator, enter the following: CF0 = -20, CF1 = 10, CF2 = 9, CF3 = 6, I/YR = 5; NPV = $2.87. Change I/YR = 5 to I/YR = 10; NPV = $1.04. Change I/YR = 10 to I/YR = 15; NPV = -$0.55. b. Using the data for Project A, enter the cash flows into a financial calculator and solve for IRRA = 11.10%. The IRR is independent of the WACC, so it doesn’t change when the WACC changes. 5 Using the data for Project B, enter the cash flows into a financial calculator and solve for IRRB = 13.18%. Again, the IRR is independent of the WACC, so it doesn’t change when the WACC changes. c. At a WACC = 5%, NPVA > NPVB so choose Project A. At a WACC = 10%, NPVB > NPVA so choose Project B. At a WACC = 15%, both NPVs are less than zero, so neither project would be chosen. 11-7 a. Project A: CF0 = -6000; CF1-5 = 2000; I/YR = 14. Solve for NPVA = $866.16. IRRA = 19.86%. MIRR calculation: 0 | -6,000 1 | 2,000 2 | 2,000 3 | 2,000 (1.14)2 (1.14)3 (1.14) 4 4 | 2,000 1.14 5 | 2,000 2,280.00 2,599.20 2,963.09 3,377.92 13,220.21 Using a financial calculator, enter N = 5; PV = -6000; PMT = 0; FV = 13220.21; and solve for MIRRA = I/YR = 17.12%. 6 Payback calculation: 0 1 | | -6,000 2,000 Cumulative CF:-6,000 -4,000 2 | 2,000 -2,000 3 | 2,000 0 4 | 2,000 2,000 5 | 2,000 4,000 Regular PaybackA = 3 years. Discounted payback calculation: 0 1 2 3 4 5 | | | | | | -6,000 2,000 2,000 2,000 2,000 2,000 Discounted CF:-6,000 1,754.39 1,538.94 1,349.94 1,184.16 1,038.74 Cumulative CF:-6,000 -4,245.61 -2,706.67 -1,356.73 -172.57 866.17 Discounted PaybackA = 4 + $172.57/$1,038.74 = 4.17 years. Project B: CF0 = -18000; CF1-5 = 5600; I/YR = 14. Solve for NPVB = $1,255.25. IRRB = 16.80%. MIRR calculation: 0 | -18,000 1 | 5,600 2 | 5,600 3 | 5,600 4 | 5,600 1.14 (1.14)2 (1.14)3 (1.14) 4 5 | 5,600 6,384.00 7,277.76 8,296.65 9,458.18 37,016.59 Using a financial calculator, enter N = 5; PV = -18000; PMT = 0; FV = 37016.59; and solve for MIRRB = I/YR = 15.51%. Payback calculation: 0 1 2 | | | -18,000 5,600 5,600 Cumulative CF:-18,000-12,400 -6,800 3 | 5,600 -1,200 4 | 5,600 4,400 5 | 5,600 10,000 Regular PaybackB = 3 + $1,200/$5,600 = 3.21 years. 7 Discounted payback calculation: 0 1 2 3 4 5 | | | | | | -18,000 5,600 5,600 5,600 5,600 5,600 Discounted CF:-18,000 4,912.28 4,309.02 3,779.84 3,315.65 2,908.46 Cumulative CF:-18,000-13,087.72 -8,778.70 -4,998.86 -1,683.21 1,225.25 Discounted PaybackB = 4 + $1,683.21/$2,908.46 = 4.58 years. Summary of capital budgeting rules results: NPV IRR MIRR Payback Discounted payback Project A $866.16 19.86% 17.12% 3.0 years 4.17 years Project B $1,225.25 16.80% 15.51% 3.21 years 4.58 years b. If the projects are independent, both projects would be accepted since both of their NPVs are positive. c. If the projects are mutually exclusive then only one project can be accepted, so the project with the highest positive NPV is chosen. Accept Project B. d. The conflict between NPV and IRR occurs due to the difference in the size of the projects. Project B is 3 times larger than Project A. 11-10 Project A: Using a financial calculator, enter the following data: CF0 = -400; CF13 = 55; CF4-5 = 225; I/YR = 10. Solve for NPV = $30.16. Project B: Using a financial calculator, enter the following data: CF0 = -600; CF12 = 300; CF3-4 = 50; CF5 = 49; I/YR = 10. Solve for NPV = $22.80. The decision rule for mutually exclusive projects is to accept the project with the highest positive NPV. In this situation, the firm would accept Project A since NPVA = $30.16 compared to NPVB = $22.80. 11-11 Project S: Using a financial calculator, enter the following data: CF0 = -15000; CF1-5 = 4500; I/YR = 14. NPVS = $448.86. Project L: Using a financial calculator, enter the following data: CF0 = -37500; CF1-5 = 11100; I/YR = 14. NPVL = $607.20. The decision rule for mutually exclusive projects is to accept the project with the highest positive NPV. In this situation, the firm would accept Project L since NPVL = $607.20 compared to NPVS = $448.86. 8 11-12 Input the appropriate cash flows into the cash flow register, and then calculate NPV at 10% and the IRR of each of the projects: Project S: CF0 = -1000; CF1 = 900; CF2 = 250; CF3-4 = 10; I/YR = 10. Solve for NPVS = $39.14; IRRS = 13.49%. Project L: CF0 = -1000; CF1 = 0; CF2 = 250; CF3 = 400; CF4 = 800; I/YR = 10. Solve for NPVL = $53.55; IRRL = 11.74%. Since Project L has the higher NPV, it is the better project, even though its IRR is less than Project S’s IRR. The IRR of the better project is IRR L = 11.74%. 9