# ch4 - CHAPTER 4 The Value of Common Stocks Answers to...

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27 CHAPTER 4 The Value of Common Stocks Answers to Practice Questions 1. Newspaper exercise, answers will vary 2. The value of a share is the discounted value of all expected future dividends. Even if the investor plans to hold a stock for only 5 years, for example, then, at the time that the investor plans to sell the stock, it will be worth the discounted value of all expected dividends from that point on. In fact, that is the value at which the investor expects to sell the stock. Therefore, the present value of the stock today is the present value of the expected dividend payments from years one through five plus the present value of the year five value of the stock. This latter amount is the present value today of all expected dividend payments after year five. 3. The market capitalization rate for a stock is the rate of return expected by the investor. Since all securities in an equivalent risk class must be priced to offer the same expected return, the market capitalization rate must equal the opportunity cost of capital of investing in the stock. 4. Expected Future Values Present Values Horizon Period (H) Dividend (DIV t ) Price (P t ) Cumulative Dividends Future Price Total 0 100.00 100.00 100.00 1 10.00 105.00 8.70 91.30 100.00 2 10.50 110.25 16.64 83.36 100.00 3 11.03 115.76 23.89 76.11 100.00 4 11.58 121.55 30.51 69.50 100.00 10 15.51 162.89 59.74 40.26 100.00 20 25.27 265.33 83.79 16.21 100.00 50 109.21 1,146.74 98.94 1.06 100.00 100 1,252.39 13,150.13 99.99 0.01 100.00 Assumptions 1. Dividends increase by 5% per year compounded. 2. The capitalization rate is 15%.

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28 5. a. Using the growing perpetuity formula, we have: P 0 = Div 1 /(r – g) 73 = 1.68/(r - 0.085) r = 0.108 = 10.8% b. We know that: Plowback ratio = 1.0 – payout ratio Plowback ratio = 1.0 - 0.5 = 0.5 And, we also know that: dividend growth rate = g = plowback ratio × ROE g = 0.5 × 0.12 = 0.06 = 6.0% Using this estimate of g, we have: P 0 = Div 1 /(r – g) 73 = 1.68/(r - 0.06) r = 0.083 = 8.3% 6. Using the growing perpetuity formula, we have: P 0 = Div 1 /(r – g) = 2/(0.12 - 0.04) = \$25 7. \$100.00 0.10 \$10 r DIV P 1 A = = = \$83.33 .04 0 0.10 5 g r DIV P 1 B = = = × + + + + + + = 6 7 6 6 5 5 4 4 3 3 2 2 1 1 C 1.10 1 0.10 DIV 1.10 DIV 1.10 DIV 1.10 DIV 1.10 DIV 1.10 DIV 1.10 DIV P \$104.50 1.10 1 0.10 12.44 1.10 12.44 1.10 10.37 1.10 8.64 1.10 7.20 1.10 6.00 1.10 5.00 P 6 6 5 4 3 2 1 C = × + + + + + + = At a capitalization rate of 10 percent, Stock C is the most valuable. For a capitalization rate of 7 percent, the calculations are similar. The results are: P A = \$142.86 P B = \$166.67 P C = \$156.48 Therefore, Stock B is the most valuable.
29 8. a. We know that g, the growth rate of dividends and earnings, is given by: g = plowback ratio × ROE g = 0.40 × 0.20 = 0.08 = 8.0% We know that: r = (DIV 1 /P 0 ) + g r = dividend yield + growth rate Therefore: r = 0.04 + 0.08 = 0.12 = 12.0% b. Dividend yield = 4%. Therefore: DIV 1 /P 0 = 0.04 DIV 1 = 0.04 × P 0 A plowback ratio of 0.4 implies a payout ratio of 0.6, and hence: DIV 1 /EPS 1 = 0.6 DIV 1 = 0.6 × EPS 1 Equating these two expressions for DIV 1 gives a relationship between price and earnings per share: 0.04 × P 0 = 0.6 × EPS 1 P 0 /EPS 1 = 15 Also, we know that: × = 0 0 1 P PVGO 1 r P EPS With (P 0 /EPS 1 ) = 15 and r = 0.12, the ratio of the present value of growth opportunities to price is 44.4 percent.

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