Chapter 18 - Solutions to Recommended Problem Sets

18-1 CHAPTER 18 EQUITY VALUATION MODELS 5. The required return is 9 percent. 6. The Gordon DDM uses the dividend for period ( t + 1), which would be 1.05. 8. a. b. c. The price falls in response to the more pessimistic dividend forecast. The forecast for current year earnings, however, is unchanged. Therefore, the P/E ratio falls. The lower P/E ratio is evidence of the diminished optimism concerning the firm's growth prospects. 9. a. Find the growth rate from g = b ´ ROE, with the plowback ratio b = .5 and ROE = 15%. This gives g = 7.5%. If the coming year's earnings are E 1 = $2, the coming year's dividends are D 1 = (1 - b ) ´ E 1 = $1. Therefore, P 0 = D 1 k - g = $1/(0.12-0.075) = $22.22 b. We can use the dividend discount model to forecast the stock price at the end of any future year t ; we just need the dividend for year t +1. For this question, we need D 4 which is (1+ g ) 3 ´ D 1 = (1+.075) 3 ´ $1 = $1.2423. Using the DDM, P 3 = D 4 /( k - g ) = $1.2423/(0.12-.075) = $27.61 Another way to find P 3 is to note that in the DDM with a constant dividend growth rate g , the stock price will grow at that same rate: P 3 = P 3 (1 + g ) 3 = $22.22(1.075) 3 = $27.60 (this result is $0.01 different from above because of rounding) $1.22 (1.05) 0.05 .09 9% $32.03 k ´ = + = = $1.05 $35 .08 8% ( .05) r k = Þ = - 1 0 $2 $18.18 0.16 0.05 D P k g = = = - - 1 0 $2 0.16 0.12 12% $50 D k g P g g = + = + Þ = =

18-2 14. FI Corporation: a. g = 5%; D 1 = $8; k = 10% P 0 = D 1 k - g = $8 .10 - .05 = $160 b. The dividend payout ratio is 8/12 = 2/3, so the plowback ratio is b = 1/3. The implied value of ROE on future investments is found by solving: g = b ´ ROE with g = 5% and b = 1/3. ROE = 15%. c. The price assuming ROE = k is just E 1 / k . P 0 = $12/.10 = $120. Therefore, the market is paying $40 per share ($160 - $120) for growth opportunities. 15. Nogro Corporation: a. P 0 = $10, E l = $2, b = .5, ROE = .20 k = D 1 / P 0 + g D 1 = .5 ´ $2 = $1 g = b ´ ROE = .5 ´ .2 = .1 Therefore, k = $1/$10 + .1 = .1 + .1 = .2 or 20 percent. b. Since k = ROE, the NPV of future investment opportunities is zero: PVGO = P 0 - E 1 k = 10 - 10 = 0 c. Since k = ROE, the stock price would be unaffected by cutting the dividend and investing the additional earnings. d. For the same reasoning in part c above, this will not change the stock price. 20. Duo Growth Co.: 0 1 2 3 4 5 ... Dt 1.00 1.25 1.5625 1.953125 g .25 .25 .25 .05 ... ... a. The dividend to be paid at the end of year 3 is the first installment of a dividend stream that will increase indefinitely at the constant growth rate of 5 percent. Therefore, we may use the constant growth model as of the end of year 2, and can calculate intrinsic value by adding the present value of the first two dividends plus the present value of the sales price of the stock at the end of year 2. The expected price 2 years from now is: P 2 = D 3 /( k - g ) = $1.953125/( .20 - .05 ) = $13.02 The PV of this expected price is:

18-3 13.02/1.20 2 = $9.04 The PV of expected dividends in years 1 and 2 is: 1.25 1.20 + 1.5625 1.20 2 = $2.13 Thus the current price should be $9.04 + $2.13 = $11.17. b. Expected dividend yield = D 1 / P 0 = 1.25/11.17 = .112 or 11.2% c. The expected price one year from now is the PV of P 2 and D 2 . P 1 = ( D 2 + P 2 )/1.20 = (1.5625 + 13.02)/1.20 = $12.15. d. The implied capital gain is ( P 1 - P 0 )/ P 0 = (12.15 - 11.17)/11.17 = .088 = 8.8 percent. The implied capital gains rate and the expected dividend yield sum to the market capitalization rate. This is consistent with the DDM.