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The present value of a share is equal to the stream of expected dividends per share up to some horizon date plus the expected price at this date, all discounted at the return that investors require. If the horizon date is far away, we simply say that stock price equals the present value of all future dividends per share. This is the dividend discount model.
If dividends are expected to grow forever at a constant rate g, then the expected return on the stock is equal to the dividend yield (DIV1/P0) plus the expected rate of dividend growth. The value of the stock according to this constant-growth dividend discount model is P0 = DIV1/(r - g).
How should investors interpret price-earnings ratios?
You can think of a share's value as the sum of two parts—the value of the assets in place and the present value of growth opportunities, that is, of future opportunities for the firm to invest in high-return projects. The price-earnings (P/E) ratio reflects the market's assessment of the firm's growth opportunities.
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Key Terms common stock initial public offering (IPO) secondary market dividend price-earnings (P/E) multiple book value liquidation value market-value balance sheet dividend discount model constant-growth dividend discount model payout ratio plowback ratio present value of growth opportunities (PVGO) sustainable growth rate
Quiz
1. Dividend Discount Model. Amazon.com has never paid a dividend, but its share price is $66 and the market value of its stock is $22 billion. Does this invalidate the dividend discount model?
2. Dividend Yield. Favored stock will pay a dividend this year of $2.40 per share. Its dividend yield is 8 percent. At what price is the stock selling?
3. Preferred Stock. Preferred Products has issued preferred stock with a $7 annual dividend that will be paid in perpetuity.
a. If the discount rate is 12 percent, at what price should the preferred sell?
b. At what price should the stock sell 1 year from now?
c. What is the dividend yield, the capital gains yield, and the expected rate of return of the stock?
4. Constant-Growth Model. Waterworks has a dividend yield of 8 percent. If its dividend is expected to grow at a constant rate of 5 percent, what must be the expected rate of return on the company's stock?
5. Dividend Discount Model. How can we say that price equals the present value of all future dividends when many actual investors may be seeking capital gains and planning to hold their shares for only a year or two? Explain.
6. Rate of Return. Steady As She Goes, Inc., will pay a year-end dividend of $2.50 per share. Investors expect the dividend to grow at a rate of 4 percent indefinitely.
a. If the stock currently sells for $25 per share, what is the expected rate of return on the stock?
b. If the expected rate of return on the stock is 16.5 percent, what is the stock price?
7. Dividend Yield. BMM Industries pays a dividend of $2 per quarter. The dividend yield on its stock is reported at 4.8 percent. What price is the stock selling at?
8. Stock Values. Integrated Potato Chips paid a $1 per share dividend yesterday. You expect the dividend to grow steadily at a rate of 4 percent per year.
a. What is the expected dividend in each of the next 3 years?
b. If the discount rate for the stock is 12 percent, at what price will the stock sell?
c. What is the expected stock price 3 years from now?
d. If you buy the stock and plan to hold it for 3 years, what payments will you receive? What is the present value of those payments? Compare your answer to (b).
9. Constant-Growth Model. A stock sells for $40. The next dividend will be $4 per share. If the rate of return earned on reinvested funds is 15 percent and the company reinvests 40 percent of earnings in the firm, what must be the discount rate?
10. Constant-Growth Model. Gentleman Gym just paid its annual dividend of $2 per share, and it is widely expected that the dividend will increase by 5 percent per year indefinitely.
a. What price should the stock sell at? The discount rate is 15 percent.
b. How would your answer change if the discount rate were only 12 percent? Why does the answer change?
11. Constant-Growth Model. Arts and Crafts, Inc., will pay a dividend of $5 per share in 1 year. It sells at $50 a share, and firms in the same industry provide an expected rate of return of 14 percent. What must be the expected growth rate of the company's dividends?
12. Constant-Growth Model. Eastern Electric currently pays a dividend of about $1.64 per share and sells for $27 a share.
a. If investors believe the growth rate of dividends is 3 percent per year, what rate of return do they expect to earn on the stock?
b. If investors' required rate of return is 10 percent, what must be the growth rate they expect of the firm?
c. If the sustainable growth rate is 5 percent, and the plowback ratio is .4, what must be the rate of return earned by the firm on its new investments?
13. Constant-Growth Model. You believe that the Non-stick Gum Factory will pay a dividend of $2 on its common stock next year. Thereafter, you expect dividends to grow at a rate of 6 percent a year in perpetuity. If you require a return of 12 percent on your investment, how much should you be prepared to pay for the stock?
14. Negative Growth. Horse and Buggy Inc. is in a declining industry. Sales, earnings, and dividends are all shrinking at a rate of 10 percent per year.
a. If r = 15 percent and DIV1 = $3, what is the value of a share?
b. What price do you forecast for the stock next year?
c. What is the expected rate of return on the stock?
d. Can you distinguish between "bad stocks" and "bad companies"? Does the fact that the industry is declining mean that the stock is a bad buy?
15. Constant-Growth Model. Metatrend's stock will generate earnings of $5 per share this year. The discount rate for the stock is 15 percent and the rate of return on reinvested earnings also is 15 percent.
a. Find both the growth rate of dividends and the price of the stock if the company reinvests the following fraction of its earnings in the firm: (i) 0 percent; (ii) 40 percent; (iii) 60 percent.
b. Redo part (a) now assuming that the rate of return on reinvested earnings is 20 percent. What is the present value of growth opportunities for each reinvestment rate?
c. Considering your answers to parts (a) and (b), can you briefly state the difference between companies experiencing growth versus companies with growth opportunities?
16. Nonconstant Growth. You expect a share of stock to pay dividends of $1.00, $1.25, and $1.50 in each of the next 3 years. You believe the stock will sell for $20 at the end of the third year.
a. What is the stock price if the discount rate for the stock is 10 percent?
b. What is the dividend yield?
17. Constant-Growth Model. Here are data on two stocks, both of which have discount rates of 15 percent:
_Stock A Stock B
Return on equity 15% 10% Earnings per share $2.00 $1.50 Dividends per share_$1.00_$1.00
a. What are the dividend payout ratios for each firm?
b. What are the expected dividend growth rates for each firm?
c. What is the proper stock price for each firm?
18. P/E Ratios. Web Cites Research projects a rate of return of 20 percent on new projects. Management plans to plow back 30 percent of all earnings into the firm. Earnings this year will be $2 per share, and investors expect a 12 percent rate of return on the stock.
a. What is the sustainable growth rate?
b. What is the stock price?
c. What is the present value of growth opportunities?
e. What would the price and P/E ratio be if the firm paid out all earnings as dividends?
f. What do you conclude about the relationship between growth opportunities and P/E ratios?
19. Constant-Growth Model. Fincorp will pay a year-end dividend of $4.80 per share, which is expected to grow at a 4 percent rate for the indefinite future. The discount rate is 12 percent.
a. What is the stock selling for?
b. If earnings are $6.20 a share, what is the implied value of the firm's growth opportunities?
20. P/E Ratios. No-Growth Industries pays out all of its earnings as dividends. It will pay its next $4 per share dividend in a year. The discount rate is 12 percent.
a. What is the price-earnings ratio of the company?
b. What would the P/E ratio be if the discount rate were 10 percent?
21. Growth Opportunities. Stormy Weather has no attractive investment opportunities. Its return on equity equals the discount rate, which is 10 percent. Its expected earnings this year are $3 per share. Find the stock price, P/E ratio, and growth rate of dividends for plowback ratios of a. zero b. .40
22. Growth Opportunities. Trend-line Inc. has been growing at a rate of 6 percent per year and is expected to continue to do so indefinitely. The next dividend is expected to be $5 per share.
a. If the market expects a 10 percent rate of return on Trend-line, at what price must it be selling?
b. If Trend-line's earnings per share will be $8, what part of Trend-line's value is due to assets in place, and what part to growth opportunities?
23. P/E Ratios. Castles in the Sand generates a rate of return of 20 percent on its investments and maintains a plowback ratio of .30. Its earnings this year will be $2 per share. Investors expect a 12 percent rate of return on the stock.
a. Find the price and P/E ratio of the firm.
b. What happens to the P/E ratio if the plowback ratio is reduced to .20? Why?
c. Show that if plowback equals zero, the earnings-price ratio E/P falls to the expected rate of return on the stock.
24. Dividend Growth. Grandiose Growth has a dividend growth rate of 20 percent. The discount rate is 10 percent. The end-of-year dividend will be $2 per share.
a. What is the present value of the dividend to be paid in Year 1? Year 2? Year 3?
b. Could anyone rationally expect this growth rate to continue indefinitely?
25. Stock Valuation. Start-up Industries is a new firm which has raised $100 million by selling shares of stock. Management plans to earn a 24 percent rate of return on equity, which is more than the 15 percent rate of return available on comparable-risk investments. Half of all earnings will be reinvested in the firm.
a. What will be Start-up's ratio of market value to book value?
b. How would that ratio change if the firm can earn only a 10 percent rate of return on its investments?
26. Nonconstant Growth. Planned Obsolescence has a product that will be in vogue for 3 years, at which point the firm will close up shop and liquidate the assets. As a result, forecasted dividends are DIV1 = $2, DIV2 = $2.50, and DIV3 = $18. What is the stock price if the discount rate is 12 percent?
27. Nonconstant Growth. Tattletale News Corp. has been growing at a rate of 20 percent per year, and you expect this growth rate in earnings and dividends to continue for another 3 years.
a. If the last dividend paid was $2, what will the next dividend be?
b. If the discount rate is 15 percent and the steady growth rate after 3 years is 4 percent, what should the stock price be today?
28. Nonconstant Growth. Reconsider Tattletale News from the previous problem.
a. What is your prediction for the stock price in 1 year?
b. Show that the expected rate of return equals the discount rate.
29. Sustainable Growth. Computer Corp. reinvests 60 percent of its earnings in the firm. The stock sells for $50, and the next dividend will be $2.50 per share. The discount rate is 15 percent. What is the rate of return on the company's reinvested funds?
30. Nonconstant Growth. A company will pay a $1 per share dividend in 1 year. The dividend in 2 years will be $2 per share, and it is expected that dividends will grow at 5 percent per year thereafter. The expected rate of return on the stock is 12 percent.
a. What is the current price of the stock?
b. What is the expected price of the stock in a year?
c. Show that the expected return, 12 percent, equals dividend yield plus capital appreciation.
31. Nonconstant Growth. Phoenix Industries has pulled off a miraculous recovery. Four years ago it was near bankruptcy. Today, it announced a $1 per share dividend to be paid a year from now, the first dividend since the crisis. Analysts expect dividends to increase by $1 a year for another 2 years. After the third year (in which dividends are $3 per share) dividend growth is expected to settle down to a more moderate long-term growth rate of 6 percent. If the firm's investors expect to earn a return of 14 percent on this stock, what must be its price?
32. Nonconstant Growth. Compost Science, Inc. (CSI), is in the business of converting Boston's sewage sludge into fertilizer. The business is not in itself very profitable. However, to induce CSI to remain in business, the Metropolitan District Commission (MDC) has agreed to pay whatever amount is necessary to yield CSI a 10 percent return on investment. At the end of the year, CSI is expected to pay a $4 dividend. It has been reinvesting 40 percent of earnings and growing at 4 percent a year.
a. Suppose CSI continues on this growth trend. What is the expected rate of return from purchasing the stock at $100?
b. What part of the $100 price is attributable to the present value of growth opportunities?
c. Now the MDC announces a plan for CSI to treat Cambridge sewage. CSI's plant will therefore be expanded gradually over 5 years. This means that CSI will have to reinvest 80 percent of its earnings for 5 years. Starting in Year 6, however, it will again be able to pay out 60 percent of earnings. What will be CSI's stock price once this announcement is made and its consequences for CSI are known?
33. Nonconstant Growth. Better Mousetraps has come out with an improved product, and the world is beating a path to its door. As a result, the firm projects growth of 20 percent per year for 4 years. By then, other firms will have copycat technology, competition will drive down profit margins, and the sustainable growth rate will fall to 5 percent. The most recent annual dividend was DIV0 = $1.00 per share.
a. What are the expected values of DIV1, DIV2, DIV3, and DIV4?
b. What is the expected stock price 4 years from now? The discount rate is 10 percent.
c. What is the stock price today?
d. Find the dividend yield, DIV1/P0.
e. What will next year's stock price, P1, be?
f. What is the expected rate of return to an investor who buys the stock now and sells it in 1 year?
Solutions to
Self-Test
1 People's Energy's high and low prices over the past 52 weeks have been 3915/16 and 281/2 per share. Its annual dividend was $2.00 per share and its dividend yield (annual dividend as a percentage of stock price) 7.0 percent. The ratio of stock price to earnings per share, the P/E ratio, is 10. Trading volume was 68,100 shares. The highest price at which the shares traded during the day was $291/4, the lowest price was $28V8, and the closing price was $283/8, which was $5/8 lower than the previous day's closing price.
2 IBM's forecast future profitability has fallen. Thus the value of future investment opportunities has fallen relative to the value of assets in place. This happens in all growth industries sooner or later, as competition increases and profitable new investment opportunities shrink.
3 P0
1.10
$100
4 Since dividends and share price grow at 5 percent,
DIV2 = $5 X 1.05 = $5.25, DIV3 = $5 X 1.052 = $5.51 P3= $100 X 1.053 = $115.76
DIV1
DIV2 DIV3 + P3
1.10 1.102
1.103
6 The two firms have equal risk, so we can use the data for Androscoggin to find the expected return on either stock:
DIV1
P0 $100
7 We've already calculated the present value of dividends through Year 3 as $2.98. We can also forecast stock price in Year 4 as
P0 = PV (dividends through Year 3) + PV(DIV4) + PV(P4) = $2 98 +$173 +$3633 $2.98 + 1.104 + 1.104 = $2.98 + $1.18 + $24.81 = $28.97
8 a. The sustainable growth rate is g = return on equity x plowback ratio = 10% x .40 = 4%
b. First value the company. At a 60 percent payout ratio, DIV1 = $3.00 as before. Using the constant-growth model,
which is $4.17 per share less than the company's no-growth value of $41.67. In this example Blue Skies is throwing away $4.17 of potential value by investing in projects with unattractive rates of return.
c. Sure. A raider could take over the company and generate a profit of $4.17 per share just by halting all investments offering less than the 12 percent rate of return demanded by investors. This assumes the raider could buy the shares for $37.50.
rence Breezeway, the CEO of Prairie Home Stores, 1 what retirement would be like. It was almost 20 years to the day since his uncle Jacob Breezeway, Prairie Home's founder, had asked him to take responsibility for managing the company. Now it was time to spend more time riding and fishing on the old Lazy Beta Ranch.
Under Mr. Breezeway's leadership Prairie Home had grown slowly but steadily and was solidly profitable. (Table 3.7 shows earnings, dividends, and book asset values for the last 5 years.) Most of the company's supermarkets had been modernized and its brand name was well-known.
Mr. Breezeway was proud of this record, although he wished that Prairie Home could have grown more rapidly. He had passed up several opportunities to build new stores in adjacent counties. Prairie Home was still just a family company. Its com-
TABLE 3.7
Financial data for Prairie Home Stores, 2000-2004 (figures in millions)
was distributed among 15 grandchildren and nephews of Jacob Breezeway, most of whom had come to depend on generous regular dividends. The commitment to high dividend payout1 had reduced the earnings available for reinvestment and thereby constrained growth.
Mr. Breezeway believed the time had come to take Prairie Home public. Once its shares were traded in the public market, the Breezeway descendants who needed (or just wanted) more cash to spend could sell off part of their holdings. Others with more interest in the business could hold on to their shares and be rewarded by higher future earnings and stock prices.
But if Prairie Home did go public, what should its shares sell for? Mr. Breezeway worried that shares would be sold, either by Breezeway family members or by the company itself, at too low a price. One relative was about to accept a private offer for $200, the
TABLE 3.7
Financial data for Prairie Home Stores, 2000-2004 (figures in millions)
2000 |
2001 |
2002 |
2003 |
2004 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Book value, start of year |
$62.7 |
66.1 |
69.0 |
73.9 |
76.5 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Earnings |
$9.7 |
9.5 |
11.8 |
11.0 |
11.2 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Dividends |
$6.3 |
6.6 |
6.9 |
7.4 |
7.7 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Retained earnings |
$3.4 |
2.9 |
4.9 |
2.6 |
3.5 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Book value, end of year |
$66.1 |
69.0 |
73.9 |
76.5 |
1. Prairie Home Stores has 400,000 common shares. 2. The company's policy is to pay cash dividends equal to 10 percent of start-of-year book value. Notes: 1. Prairie Home Stores has 400,000 common shares. 2. The company's policy is to pay cash dividends equal to 10 percent of start-of-year book value. 1 The company traditionally paid out cash dividends equal to 10 percent of start-of-period book value. See Table 5.6. TABLE 3.8 Financial projections for Prairie Home Stores, 2005-2010 (figures in millions)
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