Stocks and Their Valuation Answer Key Part 8

103. Future stock price--constant growth Answer: e Diff: M The growth rate is the required return minus the dividend yield. g = 0.13 - 0.05 = 0.08. What is D1? 0.05 = D1/$28 D1 = $1.40. What will be the Year 8 dividend? D8 = D1 × (1 + g)7 = $1.40 × (1.08)7 = $2.399354. The Year 7 price is given by: P̂7 = D8/(ks - g) = $2.399354/0.05 = $47.99. 104. Future stock price--constant growth Answer: b Diff: M First, find D6 = $2.00(1.07)5 = $2.8051. Then, calculate ks = 0.06 + 0.05(1.2) = 0.12. It follows that: P5 = $2.8051/(0.12 - 0.07) = $56.10. 105. Future stock price--constant growth Answer: b Diff: M To find the growth rate: ks = D1/P0 + g Therefore ks - D1/P0 = g 0.12 - $2/$20 = 0.02. P̂5 we can use the following formula: To find P̂5 = D6/ks - g). We therefore need D6. D6 = D1(1 + g)5 = $2(1.02)5 = $2.208. P̂5 = D6/ks - g) = $2.208/0.12 - 0.02) = $22.08. Therefore 106. Future stock price--constant growth Answer: b Diff: M Step 1: Find the cost of equity: ks = 6% + (12% - 6%)1.4 = 14.4%. Step 2: Find the value of the stock at the end of Year 1: P̂1 = D2/(ks - g) = $1.00/(0.144 - 0.05) = $10.6383. Step 3: Find the value of the stock in Year 4: P̂4 = P̂1 (1.05)3 = $10.6383(1.05)3 = $12.3152 ≈ $12.32. 107. Future stock price--constant growth Step 1: Determine the stock’s capital gains yield, g: ks = D1/P0 + g 14% = 5% + g Answer: b Diff: M 9% = g. This is the stock’s growth rate. Step 2: Calculate the stock’s price today: P0 = D1/(ks - g) = $2.00/(0.14 - 0.09) = $40. Step 3: Calculate the stock’s price 5 years from today: P̂5 = $40 × (1.09)5 = $61.545 ≈ $61.54. If the stock price today is $40 and the capital gains yield is 9 percent, the stock price must grow by 9 percent per year for the next five years, because this stock is a constant growth stock. 108. Future stock price--constant growth Answer: e Diff: M N Step 1: Calculate the firm’s cost of equity: ks = kRF + (RPM)b = 4% + (5%)1.2 = 10%. Step 2: Calculate the firm’s stock price today: D1 P0 = ks − g $ 2 . 50 = 0 . 10 −0 . 06 $= 62 . 50 . Step 3: Find the expected stock price eight years from today: P̂N = $62.50 × (1 + g)N P̂8 = $62.50 × (1.06)8 = $99.6155 ≈ $99.62. 109. FCF model for valuing stock Step 1: Answer: a Diff: M Calculate the free cash flow amount: FCF1 =EBIT(1 - T) + Depreciation - Capital Expenditur =$400 million + $80 million - $160 million - $0 =$320 million. Step 2: Calculate the firm value today using the constant growth corporate value model: FCF1 Firm value = WACC − g $ 320 = . 10 −0 . 05 ⎛ Net ⎜ operating ± Δ ⎜ working es ⎜capital ⎝ ⎞ ⎠ $ 320 0 . 05 = $6,400 million. = This is the total firm value today. Step 3: Determine the market value of the equity and price per share: MVTotal = MVEquity + MVDebt $6,400 million = MVEquity + $1,400 million MVEquity = $5,000 million. This is today’s market value of the firm’s equity. Divide by the number of shares to find the current price per share. $5,000 million/125 million = $40.00. 110. FCF model for valuing stock Answer: b Diff: M N First, we must find the expected free cash flow to be generated next year. (Remember, there was no change in net operating working capital.) FCF1 = EBIT(1 - T) + Depreciation – Gross capital expenditures FCF1 = $800(1 - 0.4) + $75 – $255 FCF1 = $300 million. Now, we can find the value of the entire firm since there is a constant growth assumption. Value of firm = FCF1/(WACC – g) Value of firm = $300/(0.09 - 0.06) Value of firm = $10,000 million. Next, we must find the value of the firm’s equity. Value of equity = Value of firm – Value of debt and preferred stock Value of equity = $10,000 – ($900 + $500) Value of equity = $8,600 million. To find the value per share of stock, we must divide the total value of the firm’s equity by the number of shares outstanding. Value per share = Value of equity/# of shares Value per share = $8,600/200 Value per share = $43.00. 111. FCF model for valuing stock Answer: b Diff: M N Time Line: 0 | FCFs Continuing Value Total FCFs 0 10% 1 | 3,000 3,000 2 3 | | 4,000 5,000 132,500 = 4,000 137,500 5,000(1 + 0.06) 0.10 – 0.06 Enter the following data as inputs in the financial calculator: CF0 = 0; CF1 = 3000; CF2 = 4000; CF3 = 137500; I = 10; and then solve for NPV = Total value of firm = $109,338.84. So, the entire company is worth $109,338.84. This, less the market value of debt and preferred stock, which was given in the problem, leaves $109,338.84 - $25,000 = $84,338.84 as the value of the firm’s common equity. The value of its common stock is calculated as $84,338.84/1,000 shares = $84.34/share. 112. FCF model for valuing stock Answer: e Diff: M N Step 1: Calculate the firm’s free cash flows (in millions of dollars) for the next year: FCF1 = EBIT(1 - T) + Dep – Cap Exp. ± ΔNOWC = $300(1 - 0.4) + $50 – $100 – $60 = $70 million. Step 2: Calculate total firm value (TFV) today: TFV = FCF1/(WACC – g) = $70/(0.10 – 0.06) = $1,750 million. Step 3: Calculate the firm’s equity value today by subtracting today’s market value of the firm’s debt and preferred stock: MVE= TFV - MVD+P = $1,750 – $500 = $1,250 million. Step 4: Calculate the firm’s price per share today: P0= MVE/# shares = $1,250/20 = $62.50. 113. New equity and equilibrium price Answer: c Diff: M Calculate new equilibrium price and determine change: $1.00(1.06 ) D0 (1.06) = = $16.06. P0, Old = 126 - 0.06 0.066 $1.065 $1.00(1 + g New) $1.00(1.06 5) = = P0, New = 0.07 135 - 0.065 k̂s, New - g New = $15.21. Change in price = $15.21 - $16.06 = -$0.85. 114. Risk and stock price Answer: a Diff: T Calculate the required rate of return: D0 = E0(Payout ratio) = $4.00(0.40) = $1.60. $1.60(1.06 ) D0(1 + g) + g = + 0.06 = 11.65%. k̂s = $30 P0 Calculate beta: 11.65% = 8% + (5%)b; b = 0.73. Calculate the new beta: bNew = 0.73(1.5) = 1.095. Calculate the new required rate of return: ks = 8% + (5%)1.095 = 13.475% ≈ 13.48%. Calculate the new expected equilibrium stock price: $ 1 . 696 P̂0 = = $22.67. 1348 −0 . 06 Change in stock price = $22.67 - $30.00 = -$7.33. 115. Constant growth stock Calculate the initial required return and equilibrium price: ks = 0.07 + (0.08)1.5 = 0.19 = 19%. $2(1.05) + g) 0 D(1 = = $15.00. P0 = 19 - 0.05 ks - g Calculate the new required return and equilibrium growth rate: New ks = 0.07 + (0.08)1.75 = 0.21. $2(1 + g) + g; P0 = $15 (Unchanged). 0.21 = $15 Answer: c Diff: T $3.15 - $2.0 = $2g + $15g $1.15 = $17g g = 0.06765 ≈ 6.77%. 116. Supernormal growth stock Time line: ks = 10% 0 1 2 gs = 20% | | | E0 = 2.00 E1 = 2.40 P0 = ? CFt D1 = 0.48 0 Answer: c Diff: T 3 g = 20%4 Years g = 20% s s | | E2 = 2.88 E3 = 3.456 E4 = 3.73248 gn = 8% D2 = 0.576 D3 = 0.6912 D4 = 1.86624 0.48 1 . 86624 P̂3 = = 93.31 0 . 10 −0 . 08 0.576 94.003 Numerical solution: $ 0 . 48 $ 0 . 576 $ 94 . 003 + + = P0 = $ 71 . 54 . 1 . 10 ( 1 . 10 )2 ( 1 . 10 ) 3 Financial calculator solution: Inputs: CF0 = 0; CF1 = 0.48; CF2 = 0.576; CF3 = 94.003; I = 10. Output: NPV = $71.54. P0 = $71.54. 117. Nonconstant growth stock Answer: b Diff: T Time line: ks = 12% 0 1 2 3 4 5 Years g1 = 0% g1 = 0% g2 = 5% g2 = 5% | | | | | | 0.50 0.50 0.50 0.525 0.55125 0.606375 P̂ = ? 2 0 . 606375 P̂4 = = 30.319 0 . 12 −0 . 10 CFt 0 0.525 30.87025 Numerical solution: gn = 10% $ 0 . 525 $ 30 . 87025 P̂2 = $= 25 . 08 . + 2 1 . 12 ( 1 . 12 ) Financial calculator solution: Calculate the PV of the stock’s expected cash flows as of time = 2. Inputs: CF0 = 0; CF1 = 0.525; CF2 = 30.87025; I = 12. P̂2 = $25.08. Output: NPV = $25.08. 118. Nonconstant growth stock Time line: 0 1 g1 = 0% 2 ks = 15% | | | P0 = 49.87 D1 = ? D2 = D 1 Answer: e Diff: T 3 gn = 10% 4 Years | | D3 = D2(1.05)D4 = D3(1.10) D4 P̂3 = 0 . 15 0 −. 10 g2 = 5% P0 = $49.87. P̂3 = (1.05)(1.1 1 0)D . 0.15 − 0.10 (1.05)(1.1 0)D 1 D1 D1 (1.05)D 1 − 0.10 + + + 0.15 2 3 3 1 . 15 ( 1 . 15 ) (1.15) ( 1 . 15 ) = + + 0.8696D 0.7561D 15.1886D 1 1 0.6904D 1 + 1 = 17.5047D D 1 = $ 2 . 85 . 49 . 87 = 119. Nonconstant growth stock Answer: c Diff: T Use the SML equation to solve for ks: ks = 0.0625 + (0.05)(1.75) = 0.15 = 15%. Calculate dividend per share: D0 = (EPS0)(Payout ratio) = ($2.50)(0.4) = $1.00. Calculate the dividend and price stream (once the stock becomes a constant growth stock): D0 = $1.00; D1 = $1.00 × 1.25 = $1.25; D2 = $1.25 × 1.20 = $1.50; 1 D3 = $1.50 × 1.15 = $1.725; D4 = $1.725 × 1.07 = $1.84575; $ 1 . 725 ( 1 . 07 ) P̂3 = $=23 . 071875 . 0 . 15 −0 . 07 Put all the cash flows on a time line: Time line: 0 3 4 Years k1s = 15% 2 | |gs = 25%| | g|s = 20% gs = 15% 1.00 1.2500 1.5000 1.7250 1.84575 P0 = ? CFt 0 1.2500 gn = 7% 1 . 84575 − 23.071875 = 0 . 15 0 . 07 1.5000 24.796875 Finally, use the cash flow register to calculate PV: CF0 = 0; CF1 = 1.25; CF2 = 1.50; CF3 = 24.796875; I = 15%; and then solve for NPV = $18.53. 120. Stock growth rate Answer: b Diff: T ks = 10%; P0 = $50; D1 = $1.00; g4+ = ? Step 1: Draw the time line: 0 ks = 10% 1 2 | gs = 25% | | gs = 25% 1.00 1.25 P0 = 50 3 gs =| 25% 1.5625 4 | gs = 25% 1.953125 | 5 Years gn = ? Step 2: Calculate the dividends: g2-4 = 25%. D1 = $1.00. D2 = $1 × (1.25) = $1.25. D3 = $1.25 × (1.25) = $1.5625. D4 = $1.5625 × (1.25) = $1.953125. Step 3: Calculate the present value of these dividends: PVdiv = $1.00/1.10 + $1.25/(1.10)2 + $1.5625/(1.10)3 + $1.953125/(1.10)4 PVdiv = $0.9091 + $1.0331 + $1.1739 + $1.3340 PVdiv = $4.4501 ≈ $4.45. Step 4: Determine the stock’s price at t = 4: The PV of the stock at t = 4 must be the future value of the difference between today’s price and the PV of the dividends through t = 4. PV P̂4 = $50.00 - $4.45 = $45.55. FV = $45.55(1.10)4 = $66.6898. This is the price at t = 4. Step 5: Determine the constant growth rate: P̂4 = D5/(ks - g) P̂4 = [D4(1 + g)]/(ks - g) $66.6898 = [$1.953125(1 + g)]/(0.10 – g) $6.66898 – $66.6898g = $1.953125 + $1.953125g $6.66898 – $1.953125 = $68.64288g $4.7158/$68.64288 = g 6.87% = g. 121. Preferred stock value Time line: 0 1 2 EAR = 6.183% | | | PV = ? Numerical solution: Pp = = $50. Answer: d Diff: T | 3 | 4 | 5 Years FV = 5,000 Amount needed to buy 100 shares: $50(100) = $5,000. $5,000 = PV(1 + 0.06/365)5(365) $5,000 = PV(1.3498) PV = $3,704.18. Financial calculator solution: Convert the nominal interest rate to an EAR: Inputs: P/YR = 365; NOM% = 6. Output: EFF% = EAR = 6.18313%. Calculate PV of deposit required today: Inputs: N = 5; I = 6.18313; PMT = 0; FV = 5000. Output: PV = -$3,704.182 ≈ -$3,704.18. Deposit $3,704.18. Note: If the financial calculator derived EAR is expressed to five decimal places it yields a PV = -$3,704.18. 122. Firm value Time line: 0 | 0.60 P0 = ? CFt Answer: c Diff: T ks = 13% gs = 15% 1 | 0.69 0 gs = 15%2 3 gs = 15% gn =44%Years | | 0.912525 0.949026 0.949026 P̂3 = = 10.54473 0.13 - 0.04 0.7935 11.4573 | 0.7935 0.69 Calculate required rate of return: ks = 8% + (4%)1.25 = 13.0%. Calculate net income, total dividends, and D0: Net income = $1.4 million/(1 - payout ratio) = $1.4 million/0.7 = $2.0 million. Dividends = $2.0 million × 0.3 = $0.6 million. D0 = $600,000/1,000,000 shares = $0.60. Financial calculator solution: Inputs: CF0 = 0; CF1 = 0.69; CF2 = 0.7935; CF3 = 11.4573; I = 13. Output: NPV = $9.17. P0 = $9.17. tal market value = P0 × Shares outstandin g = $9.17 × 1,000,000 = $9,170,000. 123. Stock price--nonconstant growth Answer: b Diff: M N First, we must find the explicit forecasted dividends: D1 = 0.75 (0.75 × 1.25 = 0.9375) D2 = 0.9375 D3 = 1.265625 (0.9375 × 1.35 = 1.265625) D4 = 1.3415625 (1.265625 × 1.06 = 1.3415625) Now, we need to determine the terminal value of the stock in Year 3, using the Year 4 dividend: P̂3 = D4/(ks – g) P̂3 = $1.3415625/(0.10 - 0.06) P̂3 = $33.5390625. 0 . 75 0 . 9375 + + 2 1 . 10 1 . 10 ) = $0.6818 + $0.7748 + $26.1493 = $27.6059 ≈ $27.61. P0 = 265625 + $ 33 . 3 ( 1 . 10 ) 5390625 Alternatively, enter all of the dividend cash flows along with the terminal value of the stock into the cash flow register and enter the 10% cost of equity to solve for the price of the stock today: CF0 = 0; CF1 = 0.75; CF2 = 0.9375; CF3 = 1.265625 + 33.5390625 = 34.8046875; I/YR = 10; and then solve for NPV = $27.61. 124. Future stock price--constant growth Answer: c Diff: M N In 10 years, this stock will be a constant growth stock. Therefore, use the constant growth formula and find the price in Year 10. In order to find the value in Year 10, determine the dividend in Year 11: D11 = 0.75 × 1.25 × 1.35 × (1.06)8 = $2.0172. Now, calculate the stock price in Year 10: P̂10 = D11/(ks – g) P̂10 = $2.0172/(0.10 - 0.06) P̂10 = $50.43. P̂3 calculated in P̂10 : Alternatively, you could have taken the terminal value the previous question and used the constant growth rate to find P̂10 = P̂3 × (1 + g)7 P̂10 = $33.5391 × (1.06)7 P̂10 = $50.43. 125. Free cash flow Answer: b Diff: E N FCF1 = EBIT(1 - T) + Depreciation – ΔNOWC – Capital expenditures = $500,000,000 - $300,000,000 = $200,000,000. 126. FCF model for valuing stock Answer: b Diff: M N P̂3 = F ĈF3 (1.07)/(0.11 – 0.07) = [($750 - $500) Using the FCF model, (1.07)]/0.04 = $6,687.50, which is the value of the firm at t = 3 after the dividend is received. So, the value of the firm today = $200/(1.11) + $200/(1.11)2 + ($250 + $6,687.50)/(1.11)3 = $5,415.1449 million ≈ $5,415 million. This is the value of the total firm (debt, preferred stock, and equity), so the value of debt and preferred stock must be deducted to arrive at the value of the firm’s common equity. The common equity has a value of $5,415 million – $700 million = $4,715 million. So, the price/share = $4,715 million/100 million = $47.15. 127. Nonconstant growth stock Answer: b Diff: M N P̂4 = D4(1 + g)/(ks – g) = $1.75(1.06)/(0.13 – 0.06) = $26.50. P̂0 = $1.75/1.13 + $1.75/(1.13)2 + $1.75/(1.13)3 + ($1.75 + $26.50)/(1.13)4 P̂0 = $1.5487 + $1.3705 + $1.2128 + $17.3263 P̂0 = $21.4583 ≈ $21.46. 128. Future stock price--nonconstant growth 2 Answer: b Diff: M N 3 P̂1 = $1.75/1.13 + $1.75/(1.13) + ($1.75 + $26.50)/(1.13) P̂1 = $1.5487 + $1.3705 + $19.5787 P̂1 = $22.4979 ≈ $22.50.