FIN303 Answers to the 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 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 +g P0 $1.00(1.06) + 0.06 $20 $1.06 = + 0.06 = 11.30%. rs = 11.30%. $20 = 9-4 a. The 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 37.80 = 3 | 1.89 1.89 0.10 0.05 The horizon, or continuing, 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 – gFCF) = $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. Each share of common stock is worth $60, according to the corporate valuation model. 9-11 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 = $10.00(1.07)4 = $13.10796 ≈ $13.11. 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. 9-15 a. Horizon value = b. 0 | $40 (1.07) 0.13 0.07 WACC = 13% ($ 17.70) 23.49 522.10 $527.89 1/1.13 = $42.80 0.06 1 | -20 1/(1.13)2 1/(1.13)3 2 | 30 = $713.33 million. 3 | 40 gn = 7% 4 | 42.80 Vop3 = 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. c. Total valuet=0 = $527.89 million. Value of common equity = $527.89 – $100 = $427.89 million. Price per share = 9-16 $427 .89 10.00 = $42.79. The value of any asset is the present value of all future cash flows expected to be generated from the asset. Hence, if we can find the present value of the dividends during the period preceding long-run constant growth and subtract that total from the current stock price, the remaining value would be the present value of the cash flows to be received during the period of long-run constant growth. D1 = $2.00 (1.25)1 = $2.50 $2.2321 D2 = $2.00 (1.25)2 = $3.125 $2.4913 D3 = $2.00 (1.25)3 = $3.90625 PV(D1) = $2.50/(1.12)1 = PV(D2) = $3.125/(1.12)2 = PV(D3) = $3.90625/(1.12)3 = $2.7804 PV(D1 to D3) = $7.5038 Therefore, the PV of the remaining dividends is: $58.8800 – $7.5038 = $51.3762. Compounding this value forward to Year 3, we find that the value of all dividends received during constant growth is $72.18. [$51.3762(1.12)3 = $72.1799 $72.18.] Applying the constant growth formula, we can solve for the constant growth rate: P̂3 = D3(1 + g)/(rs – g) $72.18 $8.6616 – $72.18g $4.7554 0.0625 6.25% = $3.90625(1 + g)/(0.12 – g) = $3.90625 + $3.90625g = $76.08625g =g = g. 9-17 0 1 rs = 12% | | g = 5% D0 = 2.00 D1 2 | D2 3 | D3 4 | D4 P̂3 a. D1 = $2(1.05) = $2.10; D2 = $2(1.05)2 = $2.2050; D3 = $2(1.05)3 = $2.31525. b. Financial calculator solution: Input 0, 2.10, 2.2050, and 2.31525 into the cash flow register, input I/YR = 12, and solve for NPV = PV = $5.28. c. Financial calculator solution: Input 0, 0, 0, and 34.73 into the cash flow register, I/YR = 12, and solve for NPV = PV = $24.72. d. $24.72 + $5.28 = $30.00 = Maximum price you should pay for the stock. P̂0 e. f. D 0 (1 g) D1 $2.10 $30.00. rs g rs g 0.12 0.05 No. The value of the stock is not dependent upon the holding period. The value calculated in Parts a through d is the value for a 3-year holding period. It is equal to the value calculated in Part e. Any other holding period would produce the same value of P̂0 ; that is, P̂0 = $30.00. Chapter 10 10-1 rd(1 – T) = 0.12(0.65) = 7.80%. 10-2 Pp = $47.50; Dp = $3.80; rp = ? rp = Dp Pp = $3.80 = 8%. $47.50 10-3 40% Debt; 60% Common equity; rd = 9%; T = 40%; WACC = 9.96%; rs = ? 10-4 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%. 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%. Chapter 11 11-2 The regular payback method has three main flaws: (1) Dollars received in different years are all given the same weight. (2) Cash flows beyond the payback year are given no consideration whatever, regardless of how large they might be. (3) Unlike the NPV, which tells us by how much the project should increase shareholder wealth, and the IRR, which tells us how much a project yields over the cost of capital, the payback merely tells us when we get our investment back. The discounted payback corrects the first flaw, but the other two flaws still remain. 11-3 The NPV is obtained by discounting future cash flows, and the discounting process actually compounds the interest rate over time. Thus, an increase in the discount rate has a much greater impact on a cash flow in Year 5 than on a cash flow in Year 1. 11-4 Mutually exclusive projects are a set of projects in which only one of the projects can be accepted. For example, the installation of a conveyor-belt system in a warehouse and the purchase of a fleet of forklifts for the same warehouse would be mutually exclusive projects—accepting one implies rejection of the other. When choosing between mutually exclusive projects, managers should rank the projects based on the NPV decision rule. The mutually exclusive project with the highest positive NPV should be chosen. The NPV decision rule properly ranks the projects because it assumes the appropriate reinvestment rate is the cost of capital. Project X should be chosen over Project Y. Since the two projects are mutually exclusive, only one project can be accepted. The decision rule that should be used is NPV. Since Project X has the higher NPV, it should be chosen. The cost of capital used in the NPV analysis appropriately includes risk. 11-8 11-1 Financial calculator solution: Input CF0 = -52125, CF1-8 = 12000, I/YR = 12, and then solve for NPV = $7,486.68. 11-2 11-4 11-5 Financial calculator solution: Input CF0 = -52125, CF1-8 = 12000, and then solve for IRR = 16%. 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. 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. 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 14% 1 | | -6,000 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%. 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 14% 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 = $1,225.25. IRRB = 16.80%. MIRR calculation: 0 14% 1 | | -18,000 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 Regular PaybackB = 3 + $1,200/$5,600 = 3.21 years. 4 | 5,600 4,400 5 | 5,600 10,000 Discounted payback calculation: 0 14% 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:-18,000-13,087.72-8,778.70-4,998.86-1,683.211,225.25 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 is greater than 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 is greater than NPVS = $448.86. 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%.