Chapter 9 Stock Valuation Before You Go On Questions and Answers Section 9.1
1. What is NASDAQ?
The National Association of Securities Automatic Quotation (NASDAQ) system is one of the largest electronic stock markets in the world, listing over three thousand companies. It was created in 1971 by the National Assoc. of Securities Dealers
2. How do dealers differ from brokers?
A dealer differs from a broker in that a dealer d ealer takes ownership of assets and is exposed exp osed to inventory risk, while a broker only facilitates a transaction on behalf of a client. Unlike brokers, dealers are subject to capital risk, because they must finance their inventories of securities.
3. List the major stock market indexes, and explain what they tell us.
The major U.S. Stock market indexes are the Dow Jones Industrial Average, New York Stock Exchange, Standard &
Poor’s 500 Index, and NASDAQ Composite Index. A
stock
market index is essentially a listing of stocks that bear some commonality, such as being traded on the same market exchange.
They are used to measure the stock market
performance and many are used to benchmark the performance of portfolios, such as mutual funds.
4. What does the price-earnings ratio tell us? The price-earnings ratio is the firm’s current price divided by the
current earnings. When valuing a company’s stock, it is useful to look at a company’s ratio and compare it to that of similar firms. In general, a high P/E ratio means high projected earnings in the future. In addition, it also tells us how much investors are willing to p ay per dollar of earnings.
5. Why do some people view preferred stock as a special type of a bond rather than a stock?
Just like debt, preferred stock is often callable and may be convertible into common stock.
Also, like debt, it has ―credit‖ ratings that are similar to those issued to bonds. But most important, preferred stock has no voting rights and just like bonds pays fixed dividends. For these reasons, many analysts treat preferred stock as a special kind of debt rather than as an equity.
Section 9.2 1.
What is the general formula used to calculate the price of a share of o f a stock? What does it mean? The general formula developed to value a share of stock is as follows: P0
D1 D2 (1 R ) (1 R ) 2
D3 (1 R ) 3
D4 (1 R ) 4
D5 (1 R )5
It says that the price of a share of stock is the present value of all expected ex pected future dividends, or: Stock price = PV (all future cash dividends).
2. What are growth stocks, and why do they typically pay little on dividends?
Growth stocks are defined as equity in any an y company whose earnings are growing g rowing faster than the average firm and the higher growth rate is expected to continue for fo r some time. Instead of
paying dividends, these firms reinvest the earnings back into the firm to pursue pursue other highreturn investment opportunities.
Section 9.3 1.
What three different models are used to value stocks based on different dividend patterns?
Based on dividend patterns, we can use the following three models to value stock: (1) zerogrowth dividend model, (2) constant-growth dividend model, mod el, or (3) supernormal dividend growth model.
2. Explain why the growth rate g must always be less than the rate of return R.
For the valuation equation to have any meaning, the firm’s dividend growth rate g must be less than the rate of return R. It also makes intuitive sense, since no firm can in the long run grow faster than the rate of economy. econom y. If that was the case, this one firm would become 100 percent of the economy and that makes no sense.
Section 9.4
1. Why can skipping payment of a preferred dividend be a bad signal?
A preferred dividend is treated like an interest payment on debt by financial markets and investors. Although the firm will not be in default, any delay d elay or failure to pay the dividends by the firm will be treated seriously by the financial markets and make them think that the firm is in financial difficulty.
2. How is a preferred stock with a fixed maturity valued?
Any preferred stock with a defined maturity date is similar to a bond with a fixed maturity date. This similarity allows for the preferred stock to be valued in a similar fashion after making adjustments for differences in the preferred stock relative to a bond. Using Equation 8.2, developed to value bonds, the coupons are replaced by the preferred dividends and the
number of payments is adjusted to four a year to accommodate the fact that firms pay pa y the dividends every quarter and not semiannually. This results in the model represented by Equation 9.7.
Self Study Problems
9.1
Ted McKay has just bought the common stock of Ryland Corp. The company expects to grow at the following rates for the next three years: 30 percent, 25 percent, and 15 percent. Last year the company paid a dividend of $2.50. Assume a required rate of return of 10 percent. Compute the expected dividends for the next three years and also the present value of these dividends.
Solution:
Expected dividends for Ryland Corp and their present value: 0
10%
1
2
3
├─────────┼─────────┼─────────┼─────────┼──────────› D0 = $2.50
D1
g 1 = 30%
D2
D3
g 2 = 25%
D1 =
D0 × (1 + g 1) = $2.50(1 + 0.30) = $3.25
D2 =
D1 × (1 + g 2) = $3.25(1 + 0.25) = $4.06
D3 =
D2 × (1 + g 3) = $4.06(1 + 0.15) = $4.67
g 3 = 15%
Present value of the dividends = PV(D P V(D1) + PV(D2) + PV(D3) = $2.96 + $3.36 + $3.51 = $9.83 9.2
Merriweather Manufacturing Company has been growing at a rate of 6 percent for the past two years, and the CEO expects the company to continue to grow at this rate for the next several years. The company paid a dividend of $1.20 last year. If your required rate
of return is 14 percent, what is the maximum max imum price that you would be willing to pay p ay for
this company’s stock? Solution:
Present value of Merriweather stock: 0
14%
1
2
3
├─────────┼─────────┼─────────┼─────────┼──────────› D0 = $1.20
D1
D2
D3
g = 6%
D1 = D0 × (1 + g ) = $1.20(1 + 0.06) = $1.27 P0
D1 ( R g)
$15.88
$1.27 (0.14 0.06)
The maximum price you should be b e willing to pay for this stock is $15.88 .
9.3
Clarion Corp. has been selling electrical supplies for the past 20 years. The company’s product line has changed very little in the past five years, and the company does not expect to add any new items for the foreseeable future. Last year, the company paid a dividend of $4.45 to its common stockholders. The company is not expected to increase its dividends for the next several years. If your required rate o f return for such firms is 13
percent, what is the current value of this company’s stock? Solution:
Present value of Clarion Corp. stock: 0
13%
1
2
3
Year
├─────────┼─────────┼─────────┼─────────┼──────────› D0 = $4.45
g = 0%
D1
D2
D3
Since the company’s dividends are not expected to grow, D0 = D1 =D2=……..D∞ = $4.45 = D Current value of the stock
9.4
=
D/R
=
$4.45/0.13
=
$34.23
Barrymore Infotech is a fast -growing communications company. The company did not pay a dividend last year and is not expected to do so for the next two years. Last year the
company’s growth accelerated, and management expects to grow the business at a rate of 35 percent for the next five years before growth slows to a more stable growth rate of 7 percent for the next several years. In the third year, the company has forecasted a dividend payment of $1.10. Calculate the value of the company’s stock at the end of its rapid growth period (i.e., at the end of five years). The required rate of o f return for such stocks is 17 percent. What is the current value of this stock?
Solution:
Present value of Barrymore Infotech stock: 0 17% 1
2
3
4
5
6
7
8
├────┼────┼────┼────┼────┼────┼────┼────┤ D0
D1
D2
g 1 – g5 = 35%
D3
D4
D5
D6
g 6 and beyond = 7%
D0 = D1 =D2 = 0 D3 = $1.10 D4 = D3 × (1 + g 4) = $1.10(1 + 0.35) = $1.485 D5 = D4 × (1 + g 5) = $1.485(1 + 0.35) = $2.005 D6 = D5 × (1 + g 6) = $2.005(1 + 0.07) = $2.145
Value of stock at t = 5 P5: P5
D6 ( R g)
$21.45
$2.145 (0.17 0.07)
Present value of the dividends in years 1 to 5 = PV(D1) + PV(D2) + PV(D3) + PV(D4) + PV(D5) $1.10 (1.17)3
$1.485 $2.005 (1.17)4 (1.17)5
$0 $0
$0 $0 $0.69 $0.79 $0.91
$2.39
Current value of stock = PV(Dividends) + PV(P5) = $2.39 + [$21.45/(1.17)5] = $2.39 + $9.78 = $ 12.17
9.5
You are interested in buying the preferred stock of a bank that pays a dividend of $1.80 every quarter. If you discount such cash flows at 8 percent, what is the value of this stock?
Solution:
Present value of preferred bank stock: Quarterly dividend on preferred stock = D = $1.80 $1.8 0 Required rate of return = 8% Current value of stock: = Po
D R
$1.8 $1.80 04 0.08
$90.00
Critical Thinking Questions 9.1
Why does the market price of a security securit y vary from the true equilibrium price?
Let us start by first defining the market equilibrium price of a s ecurity as the price that equates the demand for a security with the supply of the security. The role of the security markets is to bring buyers and sellers together in the most efficient way such that securities are bought and sold at the true equilibrium price. In reality, however, barriers of various kinds including the geographic separation of the two parties make the market price of a security slightly different different than the true equilibrium price. The more efficient the market place, the smaller the deviation between the two.
9.2
Why are investors and managers concerned about ab out stock market efficiency?
The role of secondary markets is to bring bu yers and sellers together. Ideally, we would like security markets to be as efficient as possible. Markets are efficient when current market prices of securities traded reflect all available information relevant to the security. If this is the case, security prices will be near or at their equilibrium price. The more efficient the market, the more likely this is to happen. This makes it easier for managers to price the securities close to the equilibrium price. What investors are most concerned about is having complete complet e information
regarding a security’s current price and where that price information can be obtained. Efficient markets allow them to trade at prices that are closer to the true equilibrium price than otherwise possible. Thus, both investors who provide funds and managers man agers (firms) who raise money are concerned when high transaction costs lead to inefficient markets.
9.3
Why are common stockholders considered to be more at risk than the holders of other types of securities?
In the hierarchy of lenders of funds to a firm, common stockholders have the most to lose. In the event of a firm becoming bankrupt, b ankrupt, the law requires that creditors of different types, including bondholders, be paid off o ff first. Next, preferred stockholders are paid off. Finally, common stockholders receive their investment if any funds are stil l available.
Thus, common stockholders receive their money back last and are placed at most risk. This feature of common equity is referred to as residual claim .
9.4
Under what conditions does it make sense to use the constant-growth dividend model to value a stock?
It only makes sense to use the constant-growth con stant-growth dividend model to value the stock of a company that is already paying a dividend and when that dividend can reasonably be expected to grow at a constant co nstant rate forever in other words, the stock of a mature in company.. As discussed in the chapter, the growth rate of such a company compan y will be less than the long term rate of inflation and the real growth rate of the economy. econ omy. It must also be less than the required rate of return on the stock, R.
9.5
What does it mean when a company has a very high P/E ratio? Give examples of industries in which you believe high P/E ratios are justified.
A high P/E ratio implies that investors believe that the firm has good prospects for earnings growth in the future. In fact, they the y believe that the firm will have higher growth potential than firms with lower P/E ratios. ratios. Companies in industries that are fast growing like biotech or any hi-tech industry have high P/E ratios. In the past, firms like Cisco and an d Intel had very high P/E ratios. As these firms matured and settled to annual growth rates of 15 percent or less, their P/E ratios have declined.
9.6
Preferred stock is considered to be nonparticipating because a.
investors do not participate in the election of the firm’s directors.
b.
investors do not participate in the determination of the dividend payout policy.
c.
investors do not participate in the firm’s earnings growth.
d.
none of the above.
c.
Nonparticipating implies that the preferred dividend remains constant regardless
of any increase in the firm’s earnings. Thus, investors in a firm’s preferred stock
will not see higher dividends when the firm’s earnings increase. Nor will they see a decrease if the firm’s earnings decrease. 9.7
Explain why preferred stock is considered to be a hybrid of equity and debt securities. The law considers preferred stock as equity. Thus, holders are treated as the
firm’s
owners. Also, like common stockholders, preferred stockholders have to pay taxes on their dividend income. However, preferred stockholders do not have any voting v oting rights. In addition, they receive only a fixed dividend just like bondholders. If a firm is liquidated , then they receive a stated value (par value) similar to bondholders. Preferred stock is rated by credit rating agencies just like bonds. b onds. Some preferred issues are convertible to
the firm’s common stock just as convertible bonds. Some S ome preferred issues are not perpetual and have a fixed maturity just like bonds. Thus preferred stock is a hybrid security — like like equity in some ways and like debt security in others.
9.8
Why is stock valuation more difficult than bond valuation?
Despite the availability of mathematical models to value stocks, it is more difficult to apply valuation techniques to stocks than to bonds. First, unlike bonds, firms are not in default if dividends are not declared. This makes it difficult to determine the size and timing of the cash flows. Second, common stock, unlike bonds, does not have a fixed maturity, and hence, it is difficult to determine a terminal te rminal value unlike bonds, which have a maturity value. Next, it is easier to calculate the present value of a bond because b ecause the required rate of return is observable. In the case of o f stocks, it is rather difficult to estimate a required rate of return for many stocks and classify them into different risk groups.
9.9
You are currently thinking about investing in a stock valued at $25.00 per share. The stock recently paid a dividend of $2.25 and is expected to grow at a rate of 5 percent for the foreseeable future. You normally require a return of 14 percent on stocks of similar risk. Is the stock overpriced, underpriced, or correctly priced?
This stock is underpriced at $25. Using the constant-growth model, we can arrive at a price of $26.25 for this stock. This makes the stock underpriced, and it should be considered a good buy.
9.10
Stock A and Stock B are both priced at $50 per share. Stock A has a P/E ratio of 17, while Stock B has a P/E ratio of 24. Which is the more attractive investment, considering everything else to be the same, and why?
Stock A is the more attractive investment because it has a lower P/E ratio. The lower the P/E ratio, the larger the amount of earnings supporting the stock price. This makes Stock A a more attractive investment than Stock B.
Questions and Problems BASIC 9.1
Stock Market Index: What is a stock market index?
LO 1 Solution:
A stock market index is used to measure the performance of the stock market. These indexes reflect the value of the stocks in a particular market, such as the NYSE or the NASDAQ, or across markets and increase and decrease as the values of the stocks go up and down. Examples of stock market indexes ind exes include the Dow Jones Industrial Average,
the New York Stock Exchange Index, the Standard & Poor’s Index, and the NASDAQ Composite Index. 9.2
Stock Market Index: What is the Dow Jones Industrial Average?
LO 1
Solution:
The Dow is the most widely published stock index and reflects the value of the stocks of 30 large companies. The value of the shares of these companies represents about 20 percent of the market value of all U.S. stocks.
9.3
Stock Market Index: What does NASDAQ stand for? What is NASDAQ?
Solution:
National Association of Securities Automatic Quotation system. system. NASDAQ is one of the largest electronic stock markets in the world, listing over three thousand c ompanies.
LO 1 9.4
d oes it tell us? Dividend yield: What is the dividend yield? What does
LO1
Solution:
A dividend yield is the ratio of the annual dividend payout divided by the current market price of a stock. It tells us what percentage return we would earn from dividends dividends alone if we purchased the stock at its current cu rrent price. Alternatively, it tells us what percentage of ea ch year. the firm’s stock value is being distributed to stockholders each
9.5
Present value of dividends: Fresno Corp is a fast growing company that expects to grow
at a rate of 30 percent over ov er the next two years and then slow down to a growth rate of 18 percent for the following three years. If the last dividend paid by the company was $2.15, estimate the dividends for the next five years. Compute the present value of these the se dividends if the required rate of return is 14 percent. perce nt.
LO 4 Solution:
0
1
2
3
4
5
├───────┼────────┼───────┼────────┼───────┤ D0 = $2.15
g 1-2 1-2 = 30%;
g 3-5 3-5 = 18%;
k CS CS = 14%
D1 = D0 × (1 + g 1) = $2.15(1.30) = $2.795 D2 = D1 × (1 + g 2) = $2.795(1.30) = $3.634 D3 = D2 × (1 + g 3) = $3.634(1.18) = $4.288 D4 = D3 × (1 + g 4) = $4.288(1.18) = $5.06 D5 = D4 × (1 + g 4) = $5.06(1.18) = $5.97 PV( Dividends)
9.6
$2.795 $3.634 $4.288 $5.06 (1.14)1 (1.14)2 (1.14)3 (1.14) 4
$2.45 $2.80 $2.89 $3.00 $3.10
$14.24
$5.97 (1.14)5
Zero growth: Nynet, Inc., paid a dividend of $4.18 last year. The company does not
expect to increase its dividend i the foreseeable future. If the required rate of return is 18.5 percent, what is the current price of the stock?
LO 4 Solution:
D0 = $4.18; g = 0; P0
9.7
D R
$4.18 0.185
R = 18.5%
$22.60
Zero growth: Knight Supply Corp. has not grown for the past several years and
management expects this lack of growth to continue. The firm last paid a dividend of $3.56. If you require a rate of return of 13 percent, what is the current stock price?
LO 4 Solution:
D0 = $3.56; g = 0; P0
D R
$3.56 0.13
R = 13%
$27.39
9. 8
bu ying the stock of First National Bank. Zero growth: Ron Santana is interested in buying While the bank expects no growth in the near future, Ron is attracted by b y the dividend income. Last year, the bank paid a dividend of $5.65. If Ron Santana requires a return of 14 percent on such stocks, what is the maximum price he should be willing to pay for a
share of the bank’s stock? LO 4 Solution:
D0 = $5.65; g = 0; P0
9.9
D R
$5.65 0.14
R = 14%
$40.36
Zero growth: The current stock price of Largent, Inc., is $44.72. If the required rate of
return is 19 percent, what is the dividend paid by this firm, if this dividend is not expected to grow in the future?
LO 4
Solution:
P0 = $44.72; P0
R = 19%;
D = ?;
D R
$44.72
D
0.19 D $44.72 0.19 $8.50
9.10
Constant growth: Moriband Corp. just paid a dividend of $2.15 yesterday. The company
is expected to grow at a steady stead y rate of 5 percent for the foreseeable future. If investors in stocks of companies like Moriband require a rate of o f return of 15 percent, what should shou ld be the market price of Moriband’s stock?
LO 4
Solution:
D0 = $2.15; P0
9.11
g = 5%;
D1 R g
$2.15(1.05) 0.15 0.15 0.05 0.05
R = 15%
D0 (1 g ) R g
$22.58
growing at a constant Constant growth: Nyeil, Inc., is a consumer products firm that is growing
rate of 6.5 percent. The firm’s last dividend was $3.36. $ 3.36. If the required rate of return is 18 percent, what is the market value of this stock if the dividends grow at the same rate as the firm?
LO 4
Solution:
D0 = $3.36; P0
9.12
g = 6.5%;
D1 $3.36 .36(1.065) R g 0.18 0.065
$31.12
R = 18%
Constant growth: Reco Corp. is expected to pay a dividend of $2.25 next year. The
forecast for the stock price a year from now is $37.50. If the required rate of return is 14 percent, what is the current stock price? Assume constant growth.
LO 4 Solution:
D1 = $2.25;
P1 = $37.50;
R = 14%
P1
$37.50
D2 R g
D1 (1 g ) R g
$2.25(1 g ) 0.14 g
$37.50 .50(0. (0.14 g ) $2.25 2.25g $5.25 37.5 g
2.25 2.25g
$5.25 $2.25 37.5 g 2.25g
9.13
g
$3.00 7.55% $39.75
P0
D1 R g
$2.25 $34.88 0.14 0.14 0.07 0.0755 55
Constant growth: Proxicam, Inc., is expected to grow at a constant rate of 7 percent. If
the company’s next dividend, which will be paid in a year, is $1.15 and its current stock price is $22.35, what is the required rate of return on this stock?
LO 4
Solution:
D1 = $1.15;
P0
R
P0 = $23.00; g = 7%
D1 R g D1 P0
g
$1.15 $22.35
0.07
0.05 0.0515 15 0.07 0.07 12.15%
9.14
Preferred stock valuation: X-Centric Energy Company has issued perpetual preferred
stock with a stated (par) value of $100 and a dividend of 4.5 percent. If the required rate
of return is 8.25 percent, what is the stock’s current market price? LO 6
Solution:
D = 4.5% ($100) = $4.50; P0
9.15
D R
$4.50 0.0825
R = 8.25%
$54.55
Preferred stock valuation: The First Bank of Ellicott City has issued perpetual preferred
stock with a $100 par value. The bank pays a quarterly dividend of $1.65 on this stock. What is the current price of this preferred stock given a required rate of return of 11.6 percent?
LO 6
Solution:
Quarterly dividend = $1.65 Required rate of return = R = 11.6% 11 .6% P0
9.16
(1.65 4) 0.116
$56.90
Preferred stock: The preferred stock of Axim Corp. is currently selling at $47.13. If
your required rate of return is 12.2 percent, what wh at is the dividend paid by b y this stock?
LO 6
Solution:
P0 = $47.13; P0
$47.13
$5.75
R = 12.2% D
0.122 D $47.13 0.122
9.17
Preferred stock: Each quarter, Sirkota, Inc., pays a dividend on its perpetual preferred
stock. Today, the stock is selling at $63.37. $63.3 7. If the required rate of return for such stocks is 15.5 percent, what is the quarterly qu arterly dividend paid by this firm?
LO 6
Solution:
P0 = $63.37; P0
$63.37
$9.82
R = 15.5% D
0.155 D $63.37 0.155
Annual dividend = $9.82 Quarterly dividend = $9.82 /4 = $2.46
INTERMEDIATE 9.18
Constant growth: Kay Williams is interested in purchasing the common stock of
Reckers, Inc., which is currently priced at $37.45. The company is expected to pay a dividend of $2.58 next year and to grow at a constant rate of 7 percent. a. What should the market value of the stock be if the required rate of return is 14
percent? b. Is this a good buy? Why or why not?
LO 4
Solution:
P0
a.
D1 $2.58 R g 0.14 0.07
$36.86
The stock is overpriced and not a good buy.
b.
9.19
Constant growth: The required rate of return is 23 percent. Ninex Corp. has just paid a
dividend of $3.12 of $3.12 and is expected to grow at a constant rate of 5 percent. What is the expected price of the stock three years from now?
LO 4 Solution:
R = 23%; P3
9.20
D4 R g
D0 = $3.12;
g = 5%
D0 (1 g ) 4 R g
3.12(1.05)4 0.23 0.23 0.05 0.05
$21.07
Constant growth: Jenny Banks is interested in buying the stock of Fervan, Inc., which is
increasing at a constant rate of 6 percent. pe rcent. Last year, the firm paid a dividend of $2.65. Her required rate of return is 16 percent. What is the current value of this stock? What would be the price of the stock in year 5?
LO 4 Solution:
g = 6%,
P0
P5
D1 R g
D6 R g
D0 = $2.65,
$2.65(1.06 .06) 0.16 0.06
$2.65(1.06)6 0.16 0.06
R = 16%
$28.09
$37.59
9.21
Compan y stock and are Constant growth: You own shares of Old World DVD Company interested in selling them. With so many people downloading these days, sales, profits, and dividends at Old World have been declining 6 percent per year. The firm just paid a dividend of $1.15 per share. The required rate of return for a stock this risky is 15 percent. If dividends are expected to decline at 6 percent per year, what is a share of the stock worth today?
LO 4 Solution:
R = 15%; P0
9.22
D1 R g
D0 = $1.15;
g = -6%
D0 (1 g ) R g
$1.1 $1.15 5 (1 0.06 0.06)) 0.1 0.15 ( 0.06) .06)
$5.15
Nonconstant growth: You own a company that competes with Old World DVD
Company (in the previous problem). Instead of selling DVDs, however, howev er, your company sells music downloads from a web site. Things are going well w ell now, but you know that it is only a matter of time before someone comes up with a better way to distribute music. Your company just paid a $1.50 per share dividend and you expect to increase the dividend 10 percent next year. However, you then expect your dividend growth rate to begin going down – to 5 percent the following year, 2 percent the next year, and to -3 percent per year thereafter. Based upon these estimates, what is the value of a share of
your company’s stock? Assume that the required rate of return is 12 percent. LO 5 Solution:
g 1 = 10%, g 2 = 5%, g 3 =2%,
D1 = $1.50 × 1.1 = $1.65, $1.7672,
g = -3%,
D0 = $1.50
R = 12%
D2 = $1.65 × 1.05 = $1.7325, D3 = $1.7325 ×1.02 =
D4 = $1.7672 × 0.97 = $1.7142.
9.23
D4 $1.7142 R g 0.12 (0.03)
P3
P0
P0
P0
$1.47 $1.38 $1.26 $8.14
P0
$12.25
D1 1 R $1.6 $1.65 5 1.12
D2 (1 R) R)
2
(1.12) 2
$11.43
D3
$1.7 $1.732 325 5
(1 R) R)
3
$1.7 $1.767 672 2 (1.12)3
P3 (1 R) R )3
$11. $11.43 43 (1.12)3
compan y. Nonconstant growth: Tre-Bien, Inc., is a fast growing technology company. Management projects rapid growth of 30 percent for f or the next two years, then a growth rate of 17 percent for the following two years. After that, a constant growth rate of 8 percent is expected. The firm expects to pay its first dividend of $2.45 a year from now. If dividends will grow at the same rate as the firm and the required rate of return on stocks with similar risk is 22 percent, what is the current price of the stock?
LO 5 Solution:
g 1 = g 2 = 30%,
D1 = $2.45,
g 3 = g4 = 17%,
D2 = $2.45 × 1.30 = $3.19,
D4 = $3.73 × 1.17 = $4.36, D5 $4.71 R g 0.22 0.08
P0
P0
P0
$2.01 $2.14 .14 $2.05 .05 17.15 .15
P0
$23.35
1 R $2.45 1.22
D2 (1 R) R )2 $3.19 (1.22)2
$33.64
D3 (1 R) R )3 $3.73 (1.22)3
D4 (1 R) R )4
D1 = $2.45,
D3 = $3.19 × 1.17 = $3.73
D5 = 4.36 × 1.08 = $4.71
P4
D1
g = 8%,
P4 (1 R) R )4
($4.36 33.64) (1.22)4
R = 22%
9.24
Nonconstant growth: ProCor, a biotech firm, forecasted the following growth rates for
the next three years: 35 percent, 28 percent, and 22 percent. The company then expects to grow at a constant rate of 9 percent forever. The company paid a dividend of $1.75 last week. If the required rate of return is 20 percent, p ercent, what is the value of o f this stock?
LO 5 Solution:
g 1 = 35%; g 2 = 28%; g 3 = 22%; g 4 = 9%; D0 = $1.75; R = 20%
9.25
D1
D0 (1 g 1 ) $1.75 1. 1.35 $2.36
D2
D1 (1 g 2 ) $2.36 1. 1.28 $3.02
D3
D 2 (1 g 3 ) $3.02 1. 1.22 $3.69
D4
D3 (1 g ) $3.69 1. 1 .09 $4.02
P3
D4 $4.02 R g 0.20 0.09
P0
D1 1 R
P0
$2.36 $3.0 3.02 1.20 1.20 (1.20 1.20))2
P0
$1.97 $2.10 $2.14 $21.16
P0
$27.37
D2 (1 R) R )2
$36.57
D3 (1 R) R )3 $3.6 $3.69 9 (1.20 1.20))3
P3 (1 R) R)3
$36. $36.57 57 (1.20 1.20))3
Nonconstant growth: Revarop, Inc., is a fast- growth company that is expected to grow
at a rate of 23 percent for the next four years. It is then expected to grow at a constant rate of 6 percent. Revarop’s first dividend, of $4.25, will be paid p aid in year 3. If the required rate of return is 17 percent, what is the current value v alue of the stock if the dividends are expected to grow at the same rate as the company?
LO 5
Solution:
g 1-4 1-4 = 23%; g = 6%;
D3 = $4.25;
R = 17%
D4 = D3 × 1.23 = $4.25 × 1.23 = $5.23
9.26
D5 $5.23 (1.06) R g 0.17 0.06
P4
P0
P0
00
P0
$2.65 $29.69 $32.34
D1 1 R
D2 (1 R) R )2
$4.2 $4.25 5 (1.17)3
$50.40
D3 (1 R) R )3
D4
P4
(1 R) R )4
($5. ($5.23 23 $50. $50.48 48)) 4 (1.17)
Nonconstant growth: Quansi, Inc., management expects to pay no dividends for the next
six years. It has projected a growth rate of 25 percent for the next seven years. After seven years, the firm will grow at a constant rate of 5 percent. Its first dividend, to be paid in year 7, will be $3.25. If the required rate of return is 24 percent, what is the stock worth today?
LO 5 Solution:
g constant constant = 5%; R = 24%; D7 = $3.25; D1 – D6 = 0
9.27
P7
D8 $3.25 (1.05) R g 0.24 0.05
P0
D7 P7 (1 R ) 7
$17.96
$3.25 $ $1 17.96 (1.24) 7
$4.71
Nonconstant growth: Staggert Corp. will pay dividends of $5.00, $6.25, $4.75, and
$3.00 for the next four years. Thereafter, the company expects its dividend growth rate to be constant at 6 percent. If the required rate of return is 18.5 percent, what is the current value of the stock?
LO 5
Solution:
D1 = $5;
9.28
P0
P4
P0
P0
D1 1 R
D2 = $6.25;
D5
1.185
(1 R) R)
R g
$5
D2
2
D3 = 4.75; D3 (1 R) R)
$3 (1.06) 0.185 0.06
$6.25 (1 (1.185) 2
3
D4 = $3; g = 6%; D4
(1 R) R)
4
R = 18.5%;
P4 (1 R) R )4
$25.44
$4.75 (1.185)3
($3 $25.44) (1.185)4
$4.22 $4.45 $2.86 $14.42 $25.95
Nonconstant growth: Diaz Corp. is expected to grow rapidly at a rate of 35 percent for
the next seven years. The first dividend, to be paid three years from now, will be worth $5. After seven years, the company compan y (and the dividends it pays) will settle to a constant growth rate of 8.5 percent. What is the value of this stock with a required rate of return of 14 percent?
LO 5 Solution:
g 1-7 1-7 = 35%; D3 = $5.00; g = 8.5%; R = 14%
D1 = D2 = 0;
D3 = $5
D4 = $5 × 1.35 = $6.75 D5 = $6.75 × 1.35 = $9.11 D6 =$9.11 × 1.35 = $12.30 D7 = $12.30 × 1.35 = $16.61 D8 = $16.61 × 1.085 = $18.02
9.29
P7
D8 $18.02 $ 32 7 .64 R g 0.14 0.085
P0
D1 1 R
00
$3.3 $3.38 8 $4.0 $4.00 0 $4.7 $4.73 3 $5.6 $5.60 0 $137.58
$155.29
D2 (1 R) R )2
$5 (1.14)3
D3 (1 R) R )3
$6.75 (1.14) 4
D4 (1 R) R)4
$9.11 (1.14)5
D5 (1 R) R)5
D6 (1 R) R)6
D 7 P7 (1 R) R)7
$12.30 ($16.61 $3 $327.64) 6 (1.14) (1.14)7
Nonconstant growth: Tin-Tin Waste Management, Inc., is growing rapidly. Dividends
are expected to grow at rates of 30 percent, 35 percent, 25 percent, and 18 percent over the next four years. Thereafter, management expects dividends to grow at a constant rate of 7 percent. The stock is currently selling at $47.85 and the required rate of return is 16 percent. Compute the dividend for the current year (D0).
LO 5 Solution:
g 1 = 30%; g 2 = 35%; g 3 = 25%; g 4 = 18%; g = 7%; R = 16%; P0 = $47.85
P0
D1
(1 g1 )
1.16
D0
D2 (1 g 2 )
D0 (1.30)
$47.85
2
D3 (1 g3 )
3
D4 (1 g4 )
D0 (1.30) (1.35) (1.16) 2
D0 [1.12 1.30 1.41 1.43 17.00 .00]
$22.26
$2.15
P4 (1 g 4 )4
D0 (1.30) (1.35) (1.25)
D0 (1.30 1.30)) (1.35 1.35)) (1.25 1.25)) (1.18 1.18)) 4 (1.16)
$47.85
4
(1.16)3 D0 (2.5 (2.588 886 6) (1.07) .07) 0.16 0.16 0.07 0.07 (1.16)4
ADVANCED 9.30
Equation 9.4 shows the relation between a stock’s price and the dividend that is expected next year if dividends are expected to grow at a constant rate forever. If a firm pays all of its earnings as dividends, show how Equation 9.4 can be rearranged to calculate c alculate that
firm’s P/E ratio. What does this tell us about the factors that determine a firm’s P/E ratio? LO 4
Solution:
P0
D1 R g
E1 , where E1 is the earnings pershare next year. R g
Rearranging the formula:
P0 E1
1 R g
This formula tells us that the P/E ratio is determined by both the risk of the dividend cash flows and the rate at which they are expected to grow.
9.31
Riker Departmental Stores management has forecasted a growth rate of 40 percent for the next two years, followed by growth rates of 25 percent and 20 percent pe rcent for the following two years. It then expects growth to stabilize at a constant rate of 7.5 percent percen t forever. The firm paid a dividend of $3.50 recently. If the required rate of return is 18 percent, pe rcent, what is the current value of Riker’s stock?
LO 5 Solution:
g 1-2 1-2 = 40%; g 3 = 25%; g 4 = 20%; g = 7.5%; D0 = $3.50; R = 18%
9.32
D1
D0 (1 g 1 ) $3.50 1. 1 .40 $4.90
D2
D1 (1 g 2 ) $4.90 1. 1 .40 $6.86
D3
D2 (1 g 3 ) $6.86 1.25 $8.575
D4
D3 (1 g 4 ) $8.575 1. 1 .2 0 $ 1 0 . 2 9
D5
D4 (1 g) $10.29 1. 1.075 $11.06
P4
D5 $11.06 $105.35 R g 0.18 0.075
P0
D1 1 R
P0
$4.90 $6.86 1.18 (1.18)2
P0
$4.15 $4.93 $5.22 $5.31 $54.34
P0
$73.94
D2 (1 R ) 2
D3 (1 R ) 3
D4 (1 R) R )4
P4 (1 R) R)4
$8.575 $10.29 $105.35 (1.18) 3 (1.18)4 (1.18)4
Courtesy Bancorp issued perpetual preferred stock a few years ago. The bank pays pa ys an annual dividend of $4.27, and your required rate of return is 12.2 percent. a.
What is the value of the stock given your required rate of return?
b.
Should you buy this stock if its current market price is $34.41? Explain.
LO 4 Solution: a.
D = $4.27; P0
b.
D R
R = 12.2%
$4.27 0.122
$35.00
Since the stock is worth $35.00 but can be purchased for $34.41, you should buy this stock.
9.33
Rhea Kirby owns shares in Ryoko Corp. C orp. Currently, the market price of the stock is $36.34. Management expects dividends to grow at a constant rate of 6 percent for the
foreseeable future. Its last dividend was $3.25. Rhea’s required rate of return for such stocks is 16 percent. She wants to find out whether she should sell his shares or add to her holdings. a.
What is the value of this stock?
b.
Based on your answer above, should Rhea buy additional shares in Ryoko Corp? Why or why not?
LO 4 Solution: a. b.
P0
D1 $3.25 (1.06) R g 0.16 0.06
$34.45
No, she should not buy more shares. This stock is overpriced with the stock selling at a higher price than what it is worth. She should sell her shares.
9.34
Perry, Inc., paid a dividend of $2.50 yesterday. You are interested in investing in this company, which has forecasted a constant-growth rate of 7 percent forever. Your required rate of return is 18 percent. a.
Compute the expected dividends D1, D2, D3, and D4.
b.
Find the present value of these four dividends.
c.
What is the expected value of the stock four years from now (i.e., P4)?
d.
What is the value of the stock today toda y based on the answers an swers to parts b. and c.?
e.
Use the equation for constant growth (Equation 9.4) and compute the price of the stock today.
LO 4
Solution: a.
D0 = $2.50 g = 7%
R = 18%
D1
$2.50 (1.07) $2.67 .675
D2
$2.50 (1 (1.07)2
$2.86
D3
$2.50 (1 (1.07)3
$3.063
D4
$2.50 (1 (1.07)4
$3.277
PV( Dividends) b.
D5
P4
c.
e.
D5 R g
P0
$3.063 $3.277 (1.18)3 (1.18)4
$2.27 2.05 $1.86 $1.69
$7.87
$3.506 0.18 0.07
$31.88 (1.18) 4
$31.88
$16.44
PV( Dividends) PV ( P4 )
$7.87 $16.44
$24.31
For a constant-growth stock: P0
9.35
D4 (1 g ) $3.277 (1.07) $3.506
PV( P4 ) d.
$2.675 $2.86 (1.18)1 (1.18)2
D1 $2.675 R g 0.18 0.07
$24.32
Zweite Pharma is a fast -growing drug company. compan y. Management forecasts that in the next three years, its growth rates will be 30 percent, 28 percent, and 24 percent, respectively. Last week it paid a dividend of $1.67. After three years management expects growth to stabilize at a rate of 8 percent.. The required rate of return is 14 percent. pe rcent.
a.
Compute the dividends for each of the next three years and calculate their present value.
b.
Calculate the price of the stock at the end of year 3 when the firm settles to a constant-growth rate.
c.
What is the current price of the stock?
LO 5 Solution:
g 1 = 30%; g 2 = 28%; g 3 = 24%; g = 8%; D0 = $1.67; R = 14%
D1
D0 (1 g 1 ) $1.67 (1.30) $2.171
D2
D1 (1 g 2 ) $2.171 (1 (1.28) $2.779
D3
D2 (1 g 3 ) $2.779 (1.24) $3.446
a.
PV(Dividends)
D4
P3
1 R
D2 (1 R) R )2
D3 (1 R )3
$2.1 $2.171 71 $2.7 $2.779 79 $3.4 $3.446 46 2 1.14 (1.14) (1.14)3
$1.9 $1.90 0 $2.1 $2.14 4 $2.3 $2.33 3 $6.37
D3 (1 g ) $3.446 (1.08) $3.722
b.
D4 R g
PV( P3 ) c.
D1
P0
$3.722 0.14 0.08
$62.03 (1.14) 3
$62.03
$41.87
PV( Dividends ) PV( P3 )
$6.37 $41.87
$48.24
9.36
Triton, Inc., expects to grow at a rate of 22 percent for the next nex t five years and then settle to a constant-growth rate of 6 percent. The company recently paid a dividend of $2.35. The required rate of return is 15 percent. a.
Find the present value of the dividends during the rapid growth period if the dividends grow at the same rate as the company.
b.
What is the value of the stock at the end of year 5?
c.
What is the value of the stock today?
LO 5
Solution:
g 1-5 1-5 = 22%; g = 6%; D0 = $2.35; R = 15%
a. D1
D0 (1 g 1 ) $2.35 (1 (1.22) $2.867
D2
D1 (1 g 2 ) $2.867 (1.22) $3.498
D3
D 2 (1 g 3 ) $3.498 (1.22) $4.267
D4
D3 (1 g 4 ) $4.267 (1.22) $5.206
D5
D 4 (1 g ) $5.206 (1.22) $6.351
PV(Dividends)
D6
P5
b.
D1 1 R
D2 (1 R) R )2
D3 (1 R) R)3
R g
D4 (1 R ) 4
D5 (1 R) R )5
$2.867 $3.498 $4. $4.267 $5.206 $6.351 1.15 (1.15)2 (1.15)3 (1.15)4 (1.15)5
$2.4 $2.49 9 $2.6 $2.65 5 $2.8 $2.81 1 $2.9 $2.98 8 $3.1 $3.16 6
$14.09
D5 (1 g ) $6.351 (1 (1.06) $6.732 D6
$6.732 0.15 0.06
$74.80
PV( P5 ) P0
c.
9.37
$74.80 (1.15) 5
$37.19
PV(D PV (Div iviidend ends) PV(P5 )
$14.09 $14.09 $37.19 $37.19
$51.28
Ceebros Builders is expanding very fast and expects ex pects to grow at a rate of 25 2 5 percent for the next four years. The company recently paid a dividend of $3.60 but does not expect to pay any dividends for the next three years. In year 4, management intends to pay a $5 dividend and thereafter to increase its dividend at a constant rate of 6 percent. pe rcent. The required rate of return on such stocks is 20 percent. a.
Calculate the present value of the dividends during the fast-growth period.
b.
What is the value of the stock at the end of the fast -growth period (P4)?
c.
What is the value of the stock today?
d.
Would today’s stock value be affected by the length of time you intend to hold the stock?
LO 5
Solution: a.
g 1-4 1-4 = 25% g 5 = 6% D0 = $3.60 D4 = $5.00 R = 20%
PV( Dividends) 0 0 0
b.
c. d.
P4
$5 (1.20) 4
D5 $5.00 (1.06) R g 0.20 0.06
$37.86
$37.86 (1.20) 4
$20.67
P0
$2.41
$2.41
No, the length of the holding period has no bearing on today’s stock price.
Sample Test Problems 9.1
Mason Corp. is a manufacturer of consumer staples and has experienced no growth for the past five years while paying a dividend of $3.50 every year. The CFO expects the firm to have no growth for the foreseeable future. If the required rate of return is 10 percent, what is the price of this stock today?
Solution:
D0 = $3.50; P0
9.2
g = 0;
D $3.50 R 0.10
R = 10%
$35.00
Bucknell, Inc., recently paid a dividend of $2.10. Management forecasts dividend growth of 6 percent for the foreseeable future. What is the value of the stock today with a discount rate of 13 percent?
Solution:
D0 = $2.10;
P0
9.3
g = 6%;
D1 R g
$2.10 $2.10 (1.06) 1.06) $31.80 0.13 0.13 0.06 0.06
R = 13%
D 0 (1 g ) R g
Bradley Corp. is growing at a constant rate of 7.2 percent every year. Last week the company paid a dividend of $1.85. If dividends will grow at the same s ame rate as the firm and the required rate of return is 15 percent, what should be the stock’s price four years from now?
Solution:
D0 = $1.85;
g = 7.2%;
R = 15%
9.4
D5
D 0 (1 g ) 5
P4
D5 $2.619 R g 0.15 0.072
$1.85 (1 (1.072)5
$2.619
$33.58
Wichita Technologies is expected to grow at a rate of 35 percent for the next ne xt three years and then stabilize with annual growth of 7 percent. The company will pay no dividend for the first two years and pay a dividend of o f $1.25 in year 3. What will be the value of the
company’s stock when the company’s supernormal growth ends? What is the value of the stock today? Assume that dividends will grow at the same rate as the firm once Wichita starts paying them. The firm’s required rate of return is 12 percent.
Solution:
g 1-3 1-3 = 35%; g = 7%; D1 = D2 = $0; D3 = $1.25; R = 12%
9.5
D4
D3 (1 g ) $1.25 (1.07) $1.338
P3
D4 $1.338 R g 0.12 0.07
P0
D1 1 R
P0
$0 $0 $0
P0
$0.89 $19.05
P0
$19.94
D2 (1 R) R )2 $1.2 $1.25 5 (1.12 1.12))3
$26.76
D3 (1 R) R )3
P3 (1 R) R )3
$26. $26.76 76 (1.12 1.12))3
UNC Bancorp has issued preferred stock with no maturity date. It has a par value of $100 $10 0 and pays a quarterly dividend of o f $2.25. If the required rate of return is 8 percent, what is the value of the stock today?
Solution:
Quarterly dividend = $2.25 Required rate of return = R = 8% P0
($2. ($2.25 25 4) $112.50 0.08
