As the market matures, and Research in Motion’s growth opportunities moderate, investors may justifiably believe that Research in Motion will enjoy high future earnings and will pay divi
Trang 1Solutions to Chapter 6 Valuing Stocks
1. No The dividend discount model allows for the fact that firms may not currently
pay dividends As the market matures, and Research in Motion’s growth
opportunities moderate, investors may justifiably believe that Research in Motion will enjoy high future earnings and will pay dividends then The stock price today can still reflect the present value of the expected per share stream of dividends
2. Dividend yield = Expected dividend/Price = DIV1/P0
So: P0 = DIV1/dividend yield
P0 = $2.4/.08 = $30
3 a The typical preferred stock pays a level perpetuity of dividends The expected
dividend next year is the same as this year’s dividend, $7 Thus the dividend growth rate is zero and the price today is:
P0 = D1/r = 7/.12 = $58.33
b The expected dividend in two years is this year’s dividend, $7
P1= D2/r = 7/.12 = $58.33
c Dividend yield = $7/$58.33 = 12 =12%
Expected capital gains = 0
Expected rate of return = 12%
4 r = DIV1/P0 + g = 8% + 5% = 13%
5 The value of a 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
6 a P0 = r = + g
r = + 04 = 14 = 14%
Trang 2b P0 = 2.50/(.165 04) = $20
7 The dividend yield is defined as the annual dividend (or the annualized current
dividend) divided by the current price The current annual dividend is ($2 4) = $8 and the dividend yield is:
DIV1/P0 = 048 $8/ P0 = 048 P0 = $8/.048 = $166.7
To work with the quarterly dividend, divide the dividend yield by 4 and repeat the above steps:
Quarterly DIV/P0 = 048/4 = 012 $2/ P0 = 012 P0 = $2/.012 = $166.7
8 Weak, semi-strong, strong, fundamental, technical
9 True The search for information and insightful analysis is what 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 on for mispriced stocks
10 Two such behavioral biases are alluded to in the text
a.) Attitudes toward risk: Psychologists have observed the tendency of investors to be particularly averse to the possibility of incurring losses Consequently, when investors sustain a loss they tend to exercise excessive conservatism in their subsequent
investment decisions to avoid worsening their deficit Conversely, when investors have secured a gain they are more eager to take on risky bets because they take comfort in their cushion of profit This behavioral bias has been implicated as a cause of stock market “bubbles”: a phenomenon where market prices far exceed intrinsic firm values b.) Beliefs about probabilities: This bias is twofold Firstly, investors tend to look at previous periods and assume that future market fluctuations will follow suit This is incorrect since stock prices follow a random walk and ought to be independent of previous market movements To further compound their error, investors tend to be myopic and place greater emphasis on recent market developments; as such, they overlook valuable information from the distant past (i.e market reactions to long-term economic cycles) Secondly, investors tend to fall prey to overconfidence When investors profit in the market they wrongly attribute their success to skill rather than mere luck Such excessive optimism may cause prices to greatly deviate from intrinsic share values
11 a DIV1 = $1 1.04 = $1.04
Trang 3DIV2 = $1 1.042 = $1.0816
DIV3 = $1 1.043 = $1.1249
b P0 = DIV1/(r g) = = $13
c P3 = DIV4/(r g) = = $14.6237
d Your payments are:
Sum of PV = $13.00, the same as your answer to (b)
12 Dividend growth rate, g = return on equity × plowback ratio:
g = 15 40 = 06
r = + g = + 06 = 16 = 16%
13 a P0 = = = $21
b P0 = = $30
The lower discount rate makes the present value of future dividends higher, raising the value of the stock
c P0 = (2 x (1 – 05) / (.15 – (-.05)) = $9.50
The price is the present value of the future dividends before they are eroded by the negative growth rate
14 r = + g g = r - = 14 – = 04 = 4%
15 a r = + g = + 03 = 0926 = 9.26%
b If r = 10, then 10 = 1.64(1.03)/27 + g So g = 0374 = 3.74%
Trang 4c g = Return on equity plowback ratio
5% = Return on equity 4
Return on equity = = = 12.5%
16 P0 = DIV1/(r g)
= [$.50 x 1.06 /(.12 - 06)] / [(1 + 12) ^ 4] = $5.61
17 a P0 = DIV1/(r g) = 3/[.15 – (.10)] = 3/.25 = $12
b P1 = DIV2/(r g) = 3(1 10)/.25 = $10.80
c return = = = 150 = 15.0%
d “Bad” companies may be declining, but if the stock price already reflects this fact, the investor still can earn a fair rate of return, as we saw in part c
18 a (i) reinvest 0% of earnings: g = 0 and DIV1 = $5:
P0 = = = $33.33 (ii) reinvest 40%: g = 15% 40 = 6% and DIV1 = $5 (1 – 40) = $3:
P0 = $3/(.15 – 06) = $33.33 (iii) reinvest 60%: g = 15% 60 = 9% and DIV1 = $5 (1 – 60) = $2:
P0 = 2/(.15 – 09) = $33.33
b (i) reinvest 0%: P0 = 5/(.15 – 0) = $33.33
PVGO = $0 (ii) reinvest 40%: P0 = = $42.86
PVGO = $42.86 – $33.33 = $9.53 (iii) reinvest 60%: P0 = = $66.67
PVGO = $66.67 – $33.33 = $33.34
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
Trang 5cases, 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 PVGO is higher when the reinvestment rate is higher in this case,
since the firm is taking greater advantage of its opportunities to invest in
positive NPV projects
19 Stock exchange information
Example information: Click on the pin in the Atlantic Ocean to get the Bermuda Stock Exchange:
2007 Listed companies: 53
Types of securities traded: equity and debt
The website has limited current stock price information, unlike the Toronto Stock Exchange Click on the daily trade report and see how variable is trading volumes
20 Hollywood stock exchange
Expected result: A fun experience trading securities Students won’t have trouble
figuring out how to set up an account
21 P0 = DIV1/(r g) where g = plowback ratio x ROE
$50 = $2.50 / 15 – (.6 x ROE)
ROE = [($2.5/$50) – 15] / 60 = 16.667 %
22 a P0 = + + = 18.10
b DIV1/P0 = $1/18.10 = 0552 = 5.52%
23 Stock A Stock B
b g = ROE plowback 15% 5 = 7.5% 10% 333 = 3.33%
Note: We interpret “recent” to mean in the past The current stock price
depends on future dividends – so the next dividend must be 1 + g times higher
24 a ROE plowback ratio = 20% 3 = 6%
Trang 6b E = $2, plowback ratio = 3, r = 12, g = 06 P0 = = $23.33
c No-growth value = E/r = $2/.12 = $16.67
PVGO = P0 no-growth value = $23.33 $16.67 = $6.66
d P/E = 23.33/2 = 11.665
e If all earnings were paid as dividends, price would equal the no-growth value,
$16.67, and P/E would be 16.67/2 = 8.335
f High P/E ratios reflect expectations of high PVGO
25 a =$60
b No-growth value = E/r = $6.20/.12 = $51.67
PVGO = P0 no-growth value = $60 51.67 = $8.33
26 a Earnings = DIV1 = $4 Growth rate g = 0
P0 = = $33.33
P/E = 33.33/4 = 8.33
b If r = 10, P0 = = 40, and P/E increases to 40/4 = 10
A decrease in the required rate of return, holding dividends constant, raises the stock price and the P/E ratio
27 a Plowback ratio = 0 implies DIV1 = $3 and g = 0
Therefore, P0 = = $30
and the P/E ratio is 30/3 = 10
b Plowback ratio = 40 implies DIV1 = $3(1 – 40) = $1.80, and g = 10% 40 = 4%
Therefore P0 = $1.80/(.10 – 04) = $30
and the P/E ratio is 30/3 = 10
c Plowback ratio = 80 implies DIV1 = $3(1 – 80) = $.60, and g = 10% 80 = 8% Therefore P0 = $.60/(.10 – 08) = $30
and the P/E ratio is 30/3 = 10
Regardless of the plowback ratio, the stock price = $30 because all projects offer return on equity just equal to the opportunity cost of capital
Trang 728 a P0 = DIV1/(r g) = $5/(.10 – 06) = $125
b If Trendline followed a zero-plowback strategy, it could pay a perpetual dividend of $8 Its value would be $8/.10 = $80, and therefore, the value of assets in place is $80 The remainder of its value must be due to growth opportunities, so PVGO = $125 – $80 = $45
29 a g = 20% 30 = 6%
DIV1 = $2(1 – 30) = $1.40
P0 = DIV1/(r g) = $1.40/(.12 06) = $23.33
P/E = 23.33/2 = 11.665
b If the plowback ratio is reduced to 20, g = 20% 20 = 4%
DIV1 = $2(1 – 20) = $1.60
P0 = DIV1/(r g) = $1.60/(.12 – 04) = 20
P/E = 20/2 = 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 = $2,
P0 = $2/.12 = 16.67 and E/P = 2/16.67 = 12
DIV2 = 2(1.20) = 2.40 PV = 2.40/1.102 = 1.983
DIV3 = 2(1.20)2 = 2.88 PV = 2.88/1.103 = 2.164
b This could not continue indefinitely If it did, the stock would be worth an infinite amount Another way to think about the feasible perpetual growth rate
is to compare the company’s growth rate with the growth rate of the economy The economy grows about 3% a year To grow faster than the economy as a whole is feasible when the company is small However, to continue to grow at 20%, the company must take over other companies and eventually become the entire economy But in the long run, it still can only grow as quickly as the entire economy So it is impossible to grow at 20% in perpetuity Think about Microsoft – it has had phenomenal growth partly by acquiring other
companies and partly by growing its own businesses However, even if it were
to own all of the companies in the world, eventually its growth rate would fall
to the growth rate of the world economy We are assuming that Bill Gates is
Trang 8not able to successfully market his software to still to be discovered alien
worlds!! Finally, note too that the constant dividend growth model fails when the assumed perpetual growth rate is greater than the discount rate
31 a Book value = $100 million
First year earnings = $100 million 24 = $24 million
Dividends = Earnings (1 – plowback ratio) = $12 million
g = return on equity plowback ratio = 24 50 = 12
Market value = = $400 million
Market-to-book ratio = $400/$100 = 4
b Now g falls to 10 50 = 05, first year earnings decline to $10 million (=$100 million × 1), and dividends decline to $5 million (=$10 million × 5)
Market value = = $50 million
Market-to-book ratio = ½
This makes sense, because the firm now earns less than the required rate of return
on its investments Its project is worth less than it costs
32 P0 = [1.819/1.089] + [1.95/1.0892] + [2.08/1.0893] + [2.23/1.0894] + [2.38/1.0895] + [2.49/ (.089 - 0445)]/1.0895 = $ 44.61
33 P0 = + + = $16.59
34 a DIV1 = $2 1.20 = $2.40
b DIV1 = $2.40 DIV2 = $2.88 DIV3 = $3.456
P3 = = $32.675
P0 = + + = $28.02
c P1 = + = $29.825
Trang 9d Capital gain = P1 P0 = $29.825 $28.02 = 1.805
r = = 15 = 15%
35 a Note: If students carry at least 4 decimal places, the results will be clearer
Also, it is easier to solve the prices in reverse order
DIV1 = $.5 DIV2 = $.5 DIV3 = $.5
DIV4 = $.5 × 1.04 = $.52 DIV5 = $.5 × 1.042 = $.5408
P4 = = = $7.7257
P3 = = = $7.4286
P2 = = = $7.1429
P1 = = = $6.8855
P0 = = = $6.6536
b Year 0
Dividend yield = = = 07515
Capital gains yield = = = 03485
Dividend yield + capital gains yield = 07515 + 03485 = 11
Year 1
Dividend yield = = = 07262
Capital gains yield = = = 03738
Dividend yield + capital gains yield = 07262 + 03738 = 11
Year 2
Dividend yield = = = 0700
Capital gains yield = = = 0400
Dividend yield + capital gains yield = 07 + 04 = 11
Year 3
Trang 10Dividend yield = = = 0700
Capital gains yield = = = 0400
Dividend yield + capital gains yield = 07 + 04 = 11
Yes, each year the sum of the dividend yield and the capital gains yield equal 11 percent, the required rate of return Once the company hits constant growth rate of
4 percent, both the dividend yield and the capital gains yield also become constant
36 DIV1 = dividend payout × earnings1 = 4 × $3 = $1.2
DIV2 = dividend payout × earnings2 = 4 × $3 × 1.1 = $1.32
DIV3 = dividend payout × earnings3 = 4 × $3 × 1.12 = $1.452
DIV4 = dividend payout × earnings4 = 4 × $3 × 1.13 = $1.5972
DIV5 = dividend payout × earnings5 = 4 × $3 × 1.14 = $1.75692
P0 = + + + + × = $13.95
37 a BCE Series T is listed as BCE.PR.T-T Click on the actual share name and go to
the page describing the preferred share The dividend will be listed as $1.13 due to rounding Suppose the traded price was $15 Then the expected rate of return, or yield to maturity is:
r = = $1.1255 =.075 or 7.5%
$15
b CIBC preferred shares
Expected results: Students will quickly learn that the www.globeinvestor.com does not do a good job on preferred shares The dividend payouts are rounded to 2 digits and hence do not match well the actual dividends listed on the CIBC Web site
Example: From the CIBC website, details on CM.PR.A are found:
Ticker Symbol
CUSIP Number
Quarterly Dividend
Initial Dividend Amount
Initial Payment Date CM.PR.A 136069648 $0.33125 $0.33125 Apr 28/01
At Globeinvestor.com on the preferred stock list, click on CM.PR.A and see details
on the preferred stock The annual dividend is listed as as $1.33 The actual annual dividend is $1.325, 4 times the quarterly dividend of $.33125 If the traded stock
Trang 11price is $25.82, the expected annual rate of return on preferred share is
$1.325/$25.82 = 0.0513 or 5.13%
38 Expected Results: MB– T is Mega Brands Inc and SRF-T is Sun-Rype Foods
Emphasize to the students that they should describe the businesses of these
companies in their own words Encourage students to think about how each
company presents its products Students will look at the investor pages to see the information they provide
Example: Mega Brands manufactures and markets over 60 different toys, many of which are molded plastic Mega Blocks is a competitor to LEGO They also have paper products such as puzzles and crafts The investor page provides access to
financial reports, such as quarterly and annual reports, press releases, audio
broadcasts of analyst conference call and a link to their stock listing at the Toronto Stock Exchange
39 Before-tax rate of return:
= = = 078 = 7.8%
After-tax rate of return:
=
= = 0596 = 5.96%
40 a An individual can do crazy things, but still not affect the efficiency of 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 there is no new, relevant information released by his/her action) 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
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 Furthermore, how much risk did they take? You expect to earn a higher return if you take on more market (beta) risk
c Investor psychology is a slippery concept, more often than not used to explain price movements which the individual invoking it cannot personally explain