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Unformatted text preview: Solutions to Chapter 6 Valuing Stocks 1. No, this does not invalidate the dividend discount model. The dividend discount model allows for the fact that firms may not currently pay dividends. As the market matures, and Amazon's growth opportunities moderate, investors may justifiably believe that Amazon will enjoy high future earnings and will then pay dividends. The stock price today can still reflect the present value of the expected per share stream of dividends. Dividend yield = Dividend/Price = DIV1/P0 0.08 = 2.40/P0 P0 = $30 3. The preferred stock pays a level perpetuity of dividends. The expected dividend next year is the same as this year's dividend ($8). a. b. c. $8/0.12 = $66.67 $8/0.12 = $66.67 Dividend yield = $8/$66.67 = 0.12 =12% Capital gains yield = 0 Expected rate of return = 12% 4. r = DIV1/P0 + g = 8% + 5% = 13% 5. The value of a share of common stock equals the present value of dividends received out to the investment horizon, plus the present value of the forecast stock price at the horizon. But the stock price at the horizon date depends on expectations of dividends from that date forward. So, even if an investor plans to hold a stock for only a year for two, the price ultimately received from another investor depends on dividends to be paid after the date of purchase. Therefore, the stock's present value is the same for investors with different time horizons. a.P0 = DIV1/(r  g) $30 = $3/(r  0.04) r =0.14 = 14% b. P0 = $3/(0.165  0.04) = $24 2. 6. 61 7. The annual dividend is: $2 4 = $8 DIV1/P0 = 0.048 $8/P0 = 0.048 P0 = $8/0.048 = $166.67 8. 9. weak, semistrong, strong, fundamental, technical The statement is correct. The search for information and insightful analysis makes investor assessments of stock values as reliable as possible. Since the rewards accrue to the investors who uncover relevant information before it is reflected in stock prices, competition among these investors means that there is always an active search for mispriced stocks. a.DIV1 = $1 1.04 = $1.04 DIV2 = $1 1.042 = $1.0816 DIV3 = $1 1.043 = $1.1249 b. P0 = DIV1 $1.04 = = $13.00 r  g 0.12  0.04 10. c.P3 = d. DIV4 $1.1249 1.04 = = $14.6237 rg 0.12  0.04 Year 1 $1.04 $1.04 $0.9286 Year 2 $1.0816 $1.0816 $0.8622 Year 3 $1.1249 14.6237 $15.7486 $11.2095 Your payments will be: DIV Selling Price Total Cash Flow PV of Cash Flow Sum of PV = $13.00, the same as the answer to part (b). 11. g = return on equity plowback ratio = 0.15 0.40 = 0.06 = 6.0% 40 = 4 4 r= + 0.06 = 0.16 = 16.0% r  0.06 40 P0 = P0 = DIV1 $3 1.05 = = $31.50 r  g 0.15  0.05 $3 1.05 = $45 0.12  0.05 12. a. b. 62 The lower discount rate makes the present value of future dividends higher. 13. $50 = $5 $5 = 0.04 = 4.0% g = 0.14  0.14  g $50 14. a. b. c. r = DIV1/P0 + g = [($1.64 1.03)/27] + 0.03 = 0.0926 = 9.26% If r = 0.10, then: 0.10 = [($1.64 1.03)/27] + g g = 0.0374 = 3.74% g = return on equity plowback ratio 5% = return on equity 0.4 return on equity = 0.125 = 12.5% 15. 16. P0 = DIV1/(r  g) = $2/(0.12 0.06) = $33.33 a.P0 = DIV1/(r  g) = $3/[0.15 ( 0.10)] = $3/0.25 = $12 b. c. P1 = DIV2/(r  g) = $3(1  0.10)/0.25 = $10.80 expected rate of return = DIV1 + Capital gain $3 + ($10.80  $12) = = 0.150 = 15.0% P0 $12 d. `Bad companies' may be declining, but if the stock price already reflects this fact, the investor can still earn a fair rate of return, as we saw in part (c). 17. a.(i) reinvest 0% of earnings: g = 0 and DIV1 = $6: P0 = DIV1 $6 = = $40.00 r  g 0.15  0 (ii) reinvest 40%: g = 15% 0.40 = 6% and DIV1 = $6 (1 0.40) = $3.60 P0 = DIV1 $3.60 = = $40.00 r  g 0.15  0.06 (iii) reinvest 60%: g = 15% 0.60 = 9% and DIV1 = $6 (1 0.60) = $2.40 P0 = DIV1 $2.40 = = $40.00 r  g 0.15  0.09 63 b. (i) reinvest 0%: (ii) reinvest 40%: P0 = P0 = $6 = $40.00 PVGO = $0 0.15  0 $3.60 = $51.43 0.15  (0.2 0.40) $2.40 = $80.00 0.15  (0.2 0.60) PVGO = $51.43  $40.00 = $11.43 (iii) reinvest 60%: P0 = PVGO = $80.00  $40.00 = $40.00 c.In part (a), the return on reinvested earnings is equal to the discount rate. Therefore, the NPV of the firm's new projects is zero, and PVGO is zero in all cases, regardless of the reinvestment rate. While higher reinvestment results in higher growth rates, it does not result in a higher value of growth opportunities. This example illustrates that there is a difference between growth and growth opportunities. In part (b), the return on reinvested earnings is greater than the discount rate. Therefore, the NPV of the firm's new projects is positive, and PVGO is positive. In this case, PVGO is higher when the reinvestment rate is higher because the firm is taking greater advantage of its opportunities to invest in positive NPV projects. 18. a. b. 19. a. b. c. Payout ratio g = ROE plowback ratio Pr ice = DIV1 rg Stock A $1/$2 = 0.50 15% 0.5 = 7.5% $1 = $13.33 0.15  0.075 Stock B $1/$1.50 = 0.67 10% 0.333 = 3.33% $1 = $8.57 0.15  0.0333 P0 = $1.00 $1.25 $1.50 + $20 + + = $18.10 1.10 (1.10) 2 (1.10) 3 DIV1/P0 = $1/$18.10 = 0.0552 = 5.52% 20. a. b. c. d. g = ROE plowback ratio = 20% 0.30 = 6% E = $3, r = 0.12 P0 = $3 (1  0.30) = $35.00 0.12  0.06 Nogrowth value = E/r = $3/0.12 = $25.00 PVGO = P0  nogrowth value = $35  $25 = $10 P/E = $35/$3 = 11.667 64 e. f. If all earnings were paid as dividends, price would equal the nogrowth value ($25) and P/E would be: $25/$3 = 8.333 High P/E ratios reflect expectations of high PVGO. $2.40 = $30.00 0.12  0.04 Nogrowth value = E/r = $3.10/0.12 = $25.83 PVGO = P0  nogrowth value = $30  $25.83 = $4.17 21. a. b. 22. a.Earnings = DIV1 = $4 Growth rate = g = 0 P0 = $4 = $33.33 0.12  0 $4 = $40.00 P/E increases to: $40/$4 = 10 0.10 P/E = $33.33/$4 = 8.33 b. If r = 0.10 P0 = 23. a. Plowback ratio = 0 DIV1 = $4 and g = 0 Therefore: P0 = $4 = $40.00 P/E ratio = $40/$4 = 10 0.10  0 b. Plowback ratio = 0.40 DIV1 = $4(1 0.40) = $2.40 and g = 10% 0.40 = 4% Therefore: P0 = $2.40 = $40.00 P/E ratio = $40/$4 = 10 0.10  0.04 c. Plowback ratio = 0.80 DIV1 = $4(1 0.80) = $0.80 and g = 10% 0.80 = 8% Therefore: P0 = $0.80 = $40.00 P/E ratio = $40/$4 = 10 0.10  0.08 Regardless of the plowback ratio, the stock price = $40 because all projects offer return on equity equal to the opportunity cost of capital. 65 24. a. b. P0 = DIV1/(r  g) = $5/(0.10 0.06) = $125 If Trendline followed a zeroplowback strategy, it could pay a perpetual dividend of $8. Its value would be: $8/0.10 = $80. Therefore, the value of assets in place is $80. The remainder of its value must be due to growth opportunities, so that: PVGO = $125 $80 = $45. g = 20% 0.30 = 6% P0 = $4(1 0.30)/(0.12  0.06) = $46.67 P/E = $46.67/$4 = 11.667 25. a. b. If the plowback ratio is reduced to 0.20: g = 20% 0.20 = 4% P0 = $4(1 0.20)/(0.12 0.04) = $40 P/E = $40/$4 = 10 P/E falls because the firm's value of growth opportunities is now lower: It takes less advantage of its attractive investment opportunities. c. If the plowback ratio = 0: g = 0 and DIV1 = $4 P0 = $4/0.12 = $33.33 and E/P = $4/$33.33 = 0.12 = 12.0% 26. a. DIV1 = $2.00 DIV2 = $2(1.20) = $2.40 DIV3 = $2(1.20)2 = $2.88 PV = $2/1.10 = $1.818 PV = $2.40/1.102 = $1.983 PV = $2.88/1.103 = $2.164 b. This could not continue indefinitely. If it did, the stock would be worth an infinite amount. Book value = $200 million Earnings = $200 million 0.24 = $48 million Dividends = Earnings (1 plowback ratio) = $48 million (1 0.5) = $24 million g = return on equity plowback ratio = 0.24 0.50 = 0.12 = 12.0% Market value = $24 million = $800 million 0.15  0.12 27. a. Markettobook ratio = $800/$200 = 4 66 b. Now g falls to (0.10 0.50) = 0.05, earnings decline to $20 million, and dividends decline to $10 million. Market value = $10 million = $100 million 0.15  0.05 Markettobook ratio = This result makes sense because the firm now earns less than the required rate of return on its investments. The project is worth less than it costs. P0 = $2 $2.50 $18 + + = $16.59 2 1.12 (1.12) (1.12) 3 28. 29. a.DIV1 = $2 1.20 = $2.40 b. DIV1 = $2.40 P3 = P0 = DIV2 = $2.88 DIV3 = $3.456 $3.456 1.04 = $32.675 0.15  0.04 $2.40 $2.88 $3.456 + $32.675 + + = $28.021 1.15 (1.15) 2 (1.15) 3 $2.88 $3.456 + $32.675 + = $29.825 1.15 (1.15) 2 30. a. P0 = Capital gain = P1  P0 = $29.825  $28.021 = $1.804 b. r= $2.40 + $1.804 = 0.1500 = 15.00% 28.021 31. a.An individual can do crazy things and not affect the efficiency of financial markets. An irrational person can give assets away for free or offer to pay twice the market value. However, when the person's supply of assets or money runs out, the price will adjust back to its prior level (assuming that there is no new, relevant information released by these actions). If you are lucky enough to trade with such a person you will receive a positive gain at that investor's expense. You had better not count on this happening very often though. Fortunately, an efficient market protects irrational investors in cases less extreme than the above. Even if they trade in the market in an `irrational' manner, they can be assured of getting a fair price since the price reflects all information. 67 b. Yes, and how many people have dropped a bundle? Or more to the point, how many people have made a bundle only to lose it later? People can be lucky and some people can be very lucky; efficient markets do not preclude this possibility. c.Investor psychology is a slippery concept, more often than not used to explain price movements that the individual invoking it cannot personally explain. Even if it exists, is there any way to make money from it? If investor psychology drives up the price one day, will it do so the next day also? Or will the price drop to a `true' level? Almost no one can tell you beforehand what `investor psychology' will do. Theories based on it have no content. 32. Investments in financial markets, such as stocks or bonds, are available to all participants in the marketplace. As a result, the prices of these investments are bid up to `fair' levels, that is, prices which reflect the present value of expected cash flows. If the investment weren't zeroNPV, investors would buy or sell the asset and thereby put pressure on its price until the investment becomes a zeroNPV prospect. In contrast, investments in product markets are made by firms with various forms of protection from full competition. Such protection comes from specialized knowledge, name recognition and customer loyalty, and patent protection. In these cases, a project may be positive NPV for one firm with the knowhow to make it work, but not positive NPV for other firms. Or a project may be positive NPV, but only available to one firm because it owns a name brand or patent. In these cases, competitors are kept out of the market, and the costs of the firm's investment opportunities are not bid to levels at which NPV is reduced to zero. 33. There are several thousand mutual funds in the United States. With so many professional managers, it is no surprise that some managers will demonstrate brilliant performance over various periods of time. As an analogy, consider a contest in which 10,000 people flip a coin 20 times. It would not surprise you if someone managed to flip heads 18 out of 20 times. But it would be surprising if he could repeat that performance. Similarly, while many investors have shown excellent performance over relatively short time horizons, and have received favorable publicity for their work, far fewer have demonstrated consistency over long periods. If the firm is stable and well run, its price will reflect this information, and the stock may not be a bargain. There is a difference between a `good company' and a `good stock.' The best buys in the stock market are not necessarily the best firms; instead, you want to identify firms that are better than anyone else realizes. When the market catches up to your assessment and prices adjust, you will profit. 34. 68 35. Remember the first lesson of market efficiency: The market has no memory. Just because longterm interest rates are high relative to past levels does not mean they won't go higher still. Unless you have special information indicating that longterm rates are too high, issuing longterm bonds should be a zeroNPV transaction. Issuing shortterm debt or common stock should also be a zeroNPV transaction. The stock price will decrease. The original price reflects an anticipation of a 25% increase in earnings. The actual increase is a disappointment compared to original expectations. $2.50 $2.50 = 0.10 = 10.0% g = 0.15  0.15  g $50 36. 37. $50 = g = 0.10 = return on equity plowback ratio = return on equity 0.60 return on equity = 0.10/0.60 = 0.1667 = 16.67% DIV1 DIV2 + P2 + 1+ r (1 + r ) 2 38. a. P0 = DIV1 = $2 DIV2 = $4 P2 = P0 = b. c. DIV2 (1 + g ) $4 1.05 = = $60.00 rg 0.12  0.05 $2 $4 + $60 + = $52.806 1.12 (1.12) 2 DIV2 $4 = = $57.143 0.12  0.05 0.07 Next year: P1 = r= r= DIV1 + ( P1  P0 ) $2 + ($57.143  $52.806) = = 0.1200 = 12.00% P0 $52.806 $2 $57.143  $52.806 + = 0.1200 $52.806 $52.806 69 39. DIV1 = $1 DIV2 = $2 DIV3 = $3 g = 0.06 P3 = ($3 1.06)/(0.14 0.06) = $39.75 P0 = $1 $2 $3 + $39.75 + + = $31.27 2 1.14 (1.14) (1.14) 3 DIV1 = $4 and g = 4% Expected return = (DIV1/P0) + g = ($4/$100) + 4% = 8% b. DIV1 = Earnings (1 plowback ratio) Therefore: Earnings = DIV1/(1 plowback ratio) = $4/(1 0.4) = $6.6667 If the discount rate is 8% (the expected return on the stock), then the nogrowth value of the stock is: $6.6667/0.08 = $83.33 Therefore: PVGO =$100 $83.33 = $16.67 c. For the first 5 years: g = 10% 0.8 = 8% Thereafter: g = 10% 0.4 = 4% Year Earnings plowback DIV g 1 $6.67 0.80 $1.33 0.08 2 $7.20 0.80 $1.44 0.08 3 $7.78 0.80 $1.56 0.08 4 $8.40 0.80 $1.68 0.08 5 $9.07 0.80 $1.81 0.08 6 $9.80 0.40 $5.88 0.04 40. a. After year 6, the plowback ratio falls to 0.40 and the growth rate falls to 4 percent. [We assume g = 8% in year 5 (i.e., from t = 5 to t = 6) because the plowback ratio in year 5 is still high at b = 0.80. Notice the large jump in the dividends when the plowback ratio falls.] By year 6, the firm enters a steadygrowth phase, and the constantgrowth dividend discount model can be used to value the stock. The stock price in year 6 will be: P6 = P0 = D 6 (1 + g ) $5.88 1.04 = = $152.88 kg 0.08  0.04 $1.33 $1.44 $1.56 $1.68 $1.81 $5.88 + $152.88 + + + + + = $106.22 2 3 4 1.08 (1.08) (1.08) (1.08) (1.08) 5 (1.08) 6 610 41. a. DIV1 = 1.00 1.20 = $1.20 DIV2 = 1.00 (1.20)2 = $1.44 DIV3 = 1.00 (1.20)3 = $1.728 DIV4 = 1.00 (1.20)4 = $2.0736 P4 = DIV5/(r  g) = [DIV4 (1 + g)]/(r  g) = ($2.0736 1.05)/(0.10 0.05) = $43.5456 $1.20 $1.44 $1.728 $2.0736 + $43.5456 + + + = $34.738 1.10 (1.10) 2 (1.10) 3 (1.10) 4 b. c. d. e. P0 = DIV1/P0 = $1.20/$34.738 = 0.0345 = 3.45% Next year the price will be: P1 = $1.44 $1.728 $2.0736 + $43.5456 + + = $37.012 1.10 (1.10) 2 (1.10) 3 f. r= DIV1 + Capital Gain $1.20 + ($37.012  $34.738) = = 0.1000 = 10.00% P0 $34.738 The expected return equals the discount rate (as it should if the stock is fairly priced). 611 Solution to Minicase for Chapter 6 The goal is to value the company under both investment plans and to choose the better investment plan. The discount rate is the 11% that investors believe they can earn on similarrisk investments, not the 15% return on book equity. Return on equity is useful, however, for computing the growth rate of dividends under the rapid growth scenario. Starting in 2008, in the rapid growth scenario, twothirds of earnings will be paid out as dividends, and onethird will be reinvested. Therefore, the sustainable growth rate as of 2008 is: return on equity plowback ratio = 15% 1/3 = 5% Valuation based on past growth scenario: The firm has been growing at 5% per year. Dividends are proportional to book value and have grown at 5% annually. Dividends paid in the most recent year (2007) were $7.7 million and are projected to be $8 million next year, in 2008. The value of the firm is therefore: Value 2007 = DIV2008 $8 million = = $133.33 million rg 0.11  0.05 The value per share is: $133.33 million/400,000 = $333.33 Therefore, it is clear that Mr. Breezeway was correct in advising his relative not to sell for book value of $200 per share. Valuation based on rapid growth scenario If the firm reinvests all income for the next five years (until 2012), dividends paid in that year will reach $14 million, far greater than in the constant growth scenario. However, shareholders will have to give up dividend payments until 2012 in order to achieve this rapid growth. The value of the firm in 2012 will be: Value 2012 = DIV2013 $14.7 million = = $245 million rg 0.11  0.05 The value of the firm as of the year 2007 is the present value of this amount plus the present value of the dividend to be paid in 2012, which is projected to be $14 million. (Remember, there will be no dividend flows in the years leading up to 2012.) Value 2007 = 14 + 245 = $153.70 million 1.115 Value per share is: $153.70 million/400,000 = $384.25 Thus, it appears that the rapid growth plan is in fact preferable. If the firm follows this plan, it will be able to go public  that is, sell its shares to the public  at a higher price. 612 ...
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This note was uploaded on 04/18/2008 for the course MGMT 2320 taught by Professor Park during the Spring '08 term at Rensselaer Polytechnic Institute.
 Spring '08
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