Ch 8 Solutions

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SOLUTIONS TO END-OF-CHAPTER PROBLEMS 8-1 a. Sales_1 = $800(1 + 0.125) =$900 million Sales_2 = $900(1 + 0.05) =$945 million b. NOPAT_1 = 0.10($900) = $90.00 million NOPAT_2 = 0.10($945) = $94.50 million c. OpCap_1 = 0.80($900) = $720.00 million OpCap_2 = 0.80($945) = $756.00 million d. $94.5 – ($756 – $720) = $58.50 million 8-2 Value of operations = V op = PV of expected future free cash flow V op = FCF ( 1 + g ) WACC − g = $ 400 , 000 ( 1.05 ) 0.12 − 0.05 = $6,000,000. 8-3 The growth rate in FCF from 2023 to 2024 is g = ($750.000-$707.547)/$707.547 = 0.06. HV 2024 = V Op at 2024 = FC F 2024 (1+g L ) WACC-g L = = $15,900 million. 8-4 HV 2023 = (FCF 2023 (1 + g L ))/(WACC – g L ) = ($500(1 + 0.04))/(0.09 – 0.04) = $10,400 million. V op = FC F 2022 ( 1+WACC ) 1 + FCF 2023 ( 1+WACC ) 2 + H V 2023 ( 1+WACC ) 2 = = $9,357.80 million. 8-5 V_op $800 + ST investments $70 Total value $870 -Total debt $200 -Preferred stock $50 Intrinsic value of equity $620 Divided by # shares 10 Intrinsic stock price $62.00

8-6 D 0 = $1.50; g 1-3 = 5%; g n = 10%; D 1 through D 5 = ? D 1 = D 0 (1 + g 1 ) = $1.50(1.05) = $1.5750. D 2 = D 0 (1 + g 1 )(1 + g 2 ) = $1.50(1.05) 2 = $1.6538. D 3 = D 0 (1 + g 1 )(1 + g 2 )(1 + g 3 ) = $1.50(1.05) 3 = $1.7364. D 4 = D 0 (1 + g 1 )(1 + g 2 )(1 + g 3 )(1 + g n ) = $1.50(1.05) 3 (1.10) = $1.9101. D 5 = D 0 (1 + g 1 )(1 + g 2 )(1 + g 3 )(1 + g n ) 2 = $1.50(1.05) 3 (1.10) 2 = $2.1011. 8-7 D 1 = $1.50; g = 6%; r s = 13%; ^ P 0 = ? ^ P 0 = D 1 r s − g = $ 1.50 0.13 − 0.06 = $21.43. 8-8 P 0 = $22; D 0 = $1.20; g = 10%; ^ P 1 = ?; r ¿ s = ? r ¿ s = D 1 P 0 + g = $ 1.20 ( 1.10 ) $ 22 + 0.10 = $ 1.32 $ 22 + 0.10 = 16.00%. r ¿ s = 16.00%. ^ P 1 = D 2 /(r – g) = $1.20(1.10)(1.10)/(0.16 – 0.10) = $24.20 Alternately, since this company is in a constant growth phase, the capital gains yield is equal to the dividend growth rate, so the price is also growing at the dividend growth rate: ^ P 1 = P 0 (1 + g) = $22(1.10) = $24.20. Note that this only holds when the stock is in a constant growth phase. The price will NOT grow at the year’s dividend growth rate when the company is in a nonconstant growth situation.

8-9 0 1 2 3 | | | | D 0 = 2.00 D 1 D 2 D 3 Step 1: Calculate the required rate of return on the stock: r s = r RF + (r M - r RF )b = 7.5% + (4%)1.2 = 12.3%. Step 2: Calculate the expected dividends: D 0 = $2.00 D 1 = $2.00(1.20) = $2.40 D 2 = $2.00(1.20) 2 = $2.88 D 3 = $2.88(1.07) = $3.08 Step 3: Calculate the PV of the expected dividends: PV Div = $2.40/(1.123) + $2.88/(1.123) 2 = $2.14 + $2.28 = $4.42. Step 4: Calculate ^ P 2 : ^ P 2 = D 3 /(r s – g) = $3.082/(0.123 – 0.07) = $58.11. (Note: if you carry out the calculations for D 3 to more decimal places then ^ P 2 = $58.14 and the resulting answer is $50.53 instead of the $50.50 we report in Step 6 below.) Step 5: Calculate the PV of ^ P 2 : PV = $58.11/(1.123) 2 = $46.08. Step 6: Sum the PVs to obtain the stock’s price: ^ P 0 = $4.42 + $46.08 = $50.50. Alternatively, using a financial calculator, input the following: CF 0 = 0, CF 1 = 2.40, and CF 2 = 60.99 (2.88 + 58.11) and then enter I/YR = 12.3 to solve for NPV = $50.50. 8-10 D ps = $5.00; V ps = $50; r ps = ? r ps = D ps v ps = $ 5.00 $ 50.00 = 10%.

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8-11 a. 1. V op = $ 3 ( 1 − 0.05 ) 0.13 + 0.05 = $ 2.85 0.18 = $15.83. 2. V op = $3/0.13 = $23.08. 3. V op = $ 3 ( 1.05 ) 0.13 − 0.05 = $ 3.15 0.08 = $39.38. 4. V op = $ 3 ( 1.10 ) 0.13 − 0.10 = $ 3.30 0.03 = $110.00. 8-12 a. HV 2 = $ 108 , 000 0.12 − 0.08 = $2,700,000. b. WACC = 12% so V ops = 80,000/(1.12) + 100,000/(1.12) 2 + 2,700,000/(1.12) 2 = $ 71,428.57 79,719.39 2 ,152,423.47 $2 ,303,571.43 8-13 a. HV 3 = $ 40 ( 1.07 ) 0.13 − 0.07 = $713.33. b. 0 1 2 3 4 N | | | | |    | -20 30 40 ($ 17.70) 23.49 V op 3 = 713.33 522.10 753.33 $527.89 c. Total value t = 0 = $527.89 + $10.0 = $537.89. Value of common equity = $537.89 - $100 = $437.89. Price per share = $ 437.89 10.0 = $43.79. 8-14 ^ P 0 = D 1 r s − g = D 0 ( 1 + g ) r s − g = $ 6 [ 1 +(− 0.04 )] 0.14 −(− 0.04 )] = $ 5.76 0.18 = $32.00. g = 7% WACC = 13%

8-15 The problem asks you to determine the constant growth rate, given the following facts: P 0 = $80, D 1 = $4, and r s = 14%. Use the constant growth rate formula to calculate g L : r ¿ s = D 1 P 0 + g L 0.14 = $ 4 $ 80 + g L g L = 0.09 = 9%. 8-16 The problem asks you to determine the value of ^ P 3 , given the following facts: D 1 = $3, b = 0.8, r RF = 5.2%, RP M = 6%, and P 0 = $40. Proceed as follows: Step 1: Calculate the required rate of return: r s = r RF + (r M – r RF )b = 5.2% + (6%)0.8 = 10%. Step 2: Use the constant growth rate formula to calculate g: r ¿ s = D 1 P 0 + g 0.10 = $ 3 $ 40 + g g = 0.025 = 2.5%. Step 3: Calculate ^ P 3 : ^ P 3 = P 0 (1 + g) 3 = $40(1.025) 3 = $43.076 ≈ $43.08. Alternatively, you could calculate D 4 and then use the constant growth rate formula to solve for ^ P 3 : D 4 = D 1 (1 + g) 3 = $3.00(1.025) 3 = $3.2307. ^ P 3 = $3.2307/(0.10 – 0.025) = $43.0756  $43.08. 8-17 V ps = D ps /r ps ; therefore, r ps = D ps /V ps . a. r ps = $3.5/$30 = 11.67%. b. r ps = $3.5/$40 = 8.75%. c. r ps = $3.5/$50 = 7.00%. d. r ps = $3.5/$70 = 5.00%.

8-18 D 0 = $1, r S = 7% + 6% = 13%, g 1 = 50%, g 2 = 25%, g n = 6%. D 1 = 1.50 D 2 = 1.875 D 3 = 1.9875 HV 2 = 1.9875/(0.13 – 0.06) = 28.393 ^ P 0 = 1.50/1.13 + (1.875 + 28.393)/1.13 2 = $25.03 8-19 Calculate the dividend stream and place them on a time line. Also, calculate the price of the stock at the end of the nonconstant growth period, and include it, along with the dividend to be paid at t = 5, as CF 5 . Then, enter the cash flows as shown on the time line into the cash flow register, enter the required rate of return as I = 15, and then find the value of the stock using the NPV calculation. Be sure to enter CF 0 = 0, or else your answer will be incorrect. r s = 16% D 0 = 0; D 1 = 0, D 2 = 0, D 3 = 0.50 D 4 = 0.50(1.8) = 0.9; D 5 = 0.50(1.8) 2 = 1.62; D 6 = 0.80(1.8) 2 (1.07) = $1.7334. ^ P 0 = ? 0 1 2 3 4 5 6 | | | | | | | 0.50 0.90 1.62 0.32 19.26 0.50 20.88 9.94 $10.76 = ^ P 0 ^ P 5 = D 6 /(r s – g) = 1.7334/(0.16 – 0.07) = 19.26. This is the price of the stock at the end of Year 5. CF 0 = 0; CF 1-2 = 0; CF 3 = 0.5; CF 4 = 0.9; CF 5 = 20.88; I = 16%. With these cash flows in the CFLO register, press NPV to get the value of the stock today: NPV = $10.76. 1 .7334 0 .16 − 0 .07

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8-20 a. V ps = D ps r ps = $ 10 0.08 = $125. b. V ps = $ 10 0.12 = $83.33. 8-21 a. g = $1.1449/$1.07 – 1.0 = 7%. Calculator solution: Input N = 1, PV = -1.07, PMT = 0, FV = 1.1449, I/YR = ? I = 7.00%. b. $1.07/$21.40 = 5%. c. r ¿ s = D 1 /P 0 + g = $1.07/$21.40 + 7% = 5% + 7% = 12%.

8-22 0 g=6% 1 2 3 4 | | | | | D 0 = 1.50 D 1 D 2 D 3 D 4 ^ P 3 a. D 1 = $1.5(1.06) = $1.59. D 2 = $1.50(1.06) 2 = $1.69. D 3 = $1.5(1.06) 3 = $1.79. b. PV = $1.59(0.8850) + $1.69(0.7831) + $1.79(0.6930) = $3.97. Calculator solution: Input 0, 1.59, 1.69, and 1.79 into the cash flow register, input I/YR = 13, PV = ? PV = $3.97. c. $27.05(0.6930) = $18.74. Calculator solution: Input 0, 0, 0, and 27.05 into the cash flow register, I/YR = 13, PV = ? PV = $18.74. d. $18.74 + $3.97 = $22.71 = Maximum price you should pay for the stock. (rounding differences may give you $22.72.) e. ^ P 0 = D 0 ( 1 + g ) r s − g = D 1 r s − g = $ 1.59 0.13 − 0.06 = $22.71. f. 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 except for a small rounding error. Any other holding period would produce the same value of ^ P 0 ; that is, ^ P 0 = $22.71. 8-23 a. End of Year: 0 1 2 3 4 5 6 | | | | | | | FCF 0 = 1.75 FCF 1 FCF 2 FCF 3 FCF 4 FCF 5 FCF 6 FCF t = FCF 0 (1 + g) t FCF 1 = $1.75(1.15) 1 = $2.0125 million. FCF 2 = $1.75(1.15) 2 = $2.3114 million. FCF 3 = $1.75(1.15) 3 = $2.6615 million. FCF 4 = $1.75(1.15) 4 = $3.0608 million. FCF 5 = $1.75(1.15) 5 = $3.5199 million. b. HV 5 = = = $52.80 million. This is the value of operations 5 years from now. = 12% g = 15% g = 5%

c. PV of HV 5 = $52.80/(1 + WACC) 5 = $52.80/(1 + 1.12) 5 = $29.96. Calculator solution: N = 5, I/YR = 12, PV = ?, PMT = 0, FV = -$52.80: PV = $29.96 d. Total PV of FCF 1 through FCF 5 = $2.0125(1 + 0.12) 1 + $2.3114(1 + 0.12) 2 + $2.6615(1 + 0.12) 3 + $3.0608(1 + 0.12) 4 + = $3.5199(1 + 0.12) 5 = $9.48 Calculator solution: Input 0, 2.0125, 2.3114, 2.6615, 3.0608, 3.5199 into the cash flow register, input I/YR = 12, PV = ? PV = $9.48. e. The total value of operations today is the sum or the PV of the horizon value and the PVs of the free cash flows from Year 1 through 5: V op,0 = PV FCF Years 1 through 5 + PV of HV 5 = $9.48 + $29.96 = $39.44 million. Calculator solution: Input 0, 2.01, 2.31, 2.66, 3.06, 56.32 (3.52 + 52.80) into the cash flow register, input I/YR = 12, PV = ? PV = $39.44.

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8-21 a. Graphical representation of the problem: Nonconstant Normal growth growth 0 1 2 3 ∞ | | | | D 0 D 1 (D 2 + ^ P 2 ) D 3 D ∞ PVD 1 PVD 2 PV ^ P 2 P 0 D 1 = D 0 (1 + g s ) = $2.50(1.30) = $3.25. D 2 = D 0 (1 + g s ) 2 = $2.50(1.30) 2 = $4.225. ^ P 2 = D 3 r s − g n = D 2 ( 1 + g n ) r s − g n = = $90.415 ^ P 0 = PV(D 1 ) + PV(D 2 ) + PV( ^ P 2 ) = D 1 ( 1 + r s ) + D 2 ( 1 + r s ) 2 + ^ P 2 ( 1 + r s ) 2 = = $78.3482. Calculator solution: Input 0, 3.25, 94.64 (4.225 + 90.415) into the cash flow register, input I/YR = 12, PV = ? PV = $78.3482. b. = = = $84.50. c. Expected dividend yield: D 1 /P 0 = $3.25/$78.3482 = 4.148%. Capital gains yield = ^ P 1 − P 0 P 0 = = 7.852%. Total return = Dividend yield + capital gains yield = 4.148% + 7.852% =12.00%. Notice that this is the same as the stock’s required return, r s .

d. Expected dividend yield: D 2 /P 1 = $4.225/$84.500 = 5.00%. Capital gains yield = = = 7.00%. Total return = Dividend yield + capital gains yield = 5.00% + 7.00% =12.00%. Notice that this is the same as the stock’s required return, r s . Notice also that Year 1 had a bigger capital gains yield because the dividend was growing faster in Year 1.

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