CHAPTER 9
VALUATION OF COMMON STOCKS
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9.1
9.2
9.3
9.4
9.5
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Objectives
To explain the theory and application of the discounted cash flow valuation method as applied to the equity of a
firm.
Outline
Reading Stock Listings
The Discounted Dividend Model
Earnings and Investment Opportunities
A Reconsideration of the Price/Earnings Multiple Approach
Does Dividend Policy Affect the Value of a Share?
Summary
The discounted cash flow (DCF) method of valuing assets consists of discounting expected future cash flows at
a risk-adjusted discount rate.
The discounted dividend model (DDM) for valuing shares of stock starts from the observation that an investor in
common stock expects a rate of return (consisting of cash dividends and price appreciation) that is equal to the
market capitalization rate. The resulting formula shows that the current price of a share is the present value of all
expected future dividends.
In the constant growth rate DDM, the growth rate of dividends is also the expected rate of price appreciation.
Growth per se does not add value to a share’s current price. What adds value is the opportunity to invest in
projects that yield a rate of return in excess of the market capitalization rate.
In a “frictionless” financial environment, where there are no taxes and no transaction costs, the wealth of
shareholders is the same no matter what dividend policy the firm adopts.
In the real world there are a number of frictions that can cause dividend policy to have an effect on the wealth of
shareholders. These include taxes, regulations, the costs of external finance, and the information content of
dividends.
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Solutions to Problems at End of Chapter
1. The DDM Corporation has just paid a cash dividend (D0) of $2 per share. It has consistently increased its
cash dividends in the past by 5% per year, and you expect it to continue to do so. You estimate that the
market capitalization rate for this stock should be 13% per year.
a. What is your estimate of the intrinsic value of a share (derived using the DDM model)?
b. Suppose that the actual price of a share is $20. By how much would you have to adjust each of the
following model parameters to “justify” this observed price:
i. The growth rate of dividends
ii. The market capitalization rate
SOLUTION:
a. P0 = D0(1+g)/(k-g) = 2(1+0.05)/(0.13-0.05) = $26.25
b. If the actual price of the share is $20, then some of our input parameters might need some adjustments:
i. Assuming all other parameters are left as given, then solving for g =
(.13 x 20 – 2)/(2+20) = 0.0273= 2.73%
ii. Similarly, solving for k = 2(1.05)/20 + 0.05 = 0.155 = 15.5%
2. The Rusty Clipper Fishing Corporation is expected to pay a cash dividend of $5 per share this year. You
estimate that the market capitalization rate for this stock should be 10% per year. If its current price is $25
per share, what can you infer about its expected growth rate of dividends?
SOLUTION:
D1 = $5; k = 10%; P0= $25
Hence g = 0.1 - 5/25 = -0.1 = -10%
3. The Constant Growth Corporation (CGC) has expected earnings per share (E1) of $5. It has a history of
paying cash dividends equal to 20% of earnings. The market capitalization rate for CGC’s stock is 15% per
year, and the expected ROE on the firm’s future investments is 17% per year? Using the constant growth
rate discounted dividend model,
a. What is the expected growth rate of dividends?
b. What is the model’s estimate of the present value of the stock?
c. If the model is right, what is the expected price of a share a year from now?
d. Suppose that the current price of a share is $50.
By how much would you have to adjust each of the following model parameters to “justify” this observed
price:
i. The expected ROE on the firm’s future investments.
ii. The market capitalization rate
iii. The dividend payout ratio.
SOLUTION:
a. g = earnings retention ratio x ROE = .8 x .17 = .136 = 13.6%
b. P0 = D1/(k-g)
D1 = .2 x $5 = $1 per share
P0 = $1/(.15 -.136) = $1/.014 = $71.43
c. The stock price grows at the same rate as dividends, i.e., 13.6% per year:
P1 = P0 x (1 + g) = $71.43 x 1.136 = $81.14
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d.
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If the market is efficient then the $50 price represents the best estimate of the stock’s true value. To “justify” this
price, one of the input parameters in the model needs to be adjusted:
i. Assuming all other parameters are correct, if we were to adjust for the ROE:
50 = D1/(k-g), where g = 0.8 x ROE
Solving for g, then for ROE: g = .15 – 1/50 = 0.13 = 13%,
hence ROE is equal to 13/0.8 = 16.25%
ii. If we were to adjust the market capitalization k then:
k = 1/50 + .136 = .156 = 15.6%
iii. Dividend payout ratio x E1 = 50 x (.15-.136) = 0.7,
hence Dividend payout ratio = 0.7/5=0.14 = 14%
4. The stock of Slogro Corporation is currently selling for $10 per share. Earnings per share in the coming
year are expected to be $2 per share. The company has a policy of paying out 60% of its earnings each year in
dividends. The rest is retained and invested in projects that earn a 20% rate of return per year. This situation
is expected to continue forever.
a. Assuming the current market price of the stock reflects its intrinsic value as computed using the constant
growth rate DDM, what rate of return do Slogro’s investors require?
b. By how much does its value exceed what it would be if all earnings were paid as dividends and nothing
were reinvested?
c. If Slogro were to cut its dividend payout ratio to 25%, what would happen to its stock price? What if
Slogro eliminated the dividend altogether?
d. Suppose that Slogro wishes to maintain its current 60% dividend payout policy but that it also wishes to
invest an amount each year equal to that year’s total earnings. All the money would be invested in
projects earning 20% per year. One way that Slogro could do so would be to issue an amount of new
stock each year equal to one-half that year’s earnings. What do you think would be the effect of this
policy on the current stock price?
SOLUTION:
a. P0 = $10, E1 = $2, b = .4, ROE = .2
k = D1/P0 + g
D1 = .6 x $2 = $1.20
g = b x ROE = .4 x .2 = .08
Therefore, k = $1.20/$10 + .08 = .12 + .08 = .2 or 20%
b. If all earnings were paid as dividends its price would be:
P0 = $2/.2 = $10
Thus, its price is the same whether it reinvests or not. This is because k = ROE.
c. Since k = ROE, the stock price would be unaffected by cutting the dividend and investing the additional
earnings.
d. Again, this should have no impact on the stock’s price since the NPV of the investments would be zero (the IRR
of those projects (20%) is equal to the investors’ required rate of return, hence the firm’s cost of capital).
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5. The Amazing.com Corporation currently pays no cash dividends, and it is not expected to for the next 5
years. Its sales have been growing at 25% per year.
a. Can you apply the constant growth rate DDM to estimate its intrinsic value? Explain.
b. It is expected to pay its first cash dividend $1 per share 5 years from now. It its market capitalization rate
is 20% and its dividends are expected to grow by 10% per year, what would you estimate its intrinsic
value to be?
c. If its current market price is $100 per share, what would you infer the expected growth rate of its future
dividends to be?
SOLUTION:
a. Yes, we can apply the DDM model even if the company doesn’t pay dividends for the first 5 years. The
company will eventually have to pay dividends in the future.
b. P4 = D5/(k-g) = 1/(.2-.1) = $10
P0 = 10/1.24 = $4.82
c. If P0 = $100 then P4 = 100 x 1.24 = 207.36 and g = 0.2 – 1/207.36 = 19.518%
6. The Digital Growth Corp. pays no cash dividends currently and is not expected to for the next 5 years. Its
latest EPS was $10, all of which was reinvested in the company. The firm’s expected ROE for the next 5 years
is 20% per year, and during this time it is expected to continue to reinvest all of its earnings. Starting 6 years
from now, the firm’s ROE on new investments is expected to fall to 15%, and the company is expected to start
paying out 40% of its earnings in cash dividends, which it will continue to do forever after. DG’s market
capitalization rate is 15% per year.
a. What is your estimate of DG’s intrinsic value per share?
b. Assuming its current market price is equal to its intrinsic value, what do you expect to happen to its price
over the next year? The year after?
c. What effect would it have on your estimate of DG’s intrinsic value if you expected DG to pay out only
20% of earnings starting in year 6?
SOLUTION:
Year
0
1
2
3
4
5
6
a.
b.
c.
Earnings per share
10
12
14.4
17.28
20.736
24.8832
27.12269
Dividends per share
0
0
0
0
0
0
10.84908
Plowback ratio
1
1
1
1
1
1
0.6
Growth rate
.2
.2
.2
.2
.2
.2
.09
P5 = D6/(k – g) = 10.849/(.15-.09) = $180.82
P0 = P5/(1+k)5 = $180.82/1.155 = $89.90
The price should rise by 15% per year until year 5 after which it will grow at the dividends’ growth rate g
(=9%).
Since ROE =k, the dividend payout ratio will have no effect on current price.
7. The 2Stage Co. just paid a dividend of $1 per share. The dividend is expected to grow at a rate of 25%
per year for the next 3 years and then to level off to 5% per year forever. You think the appropriate market
capitalization rate is 20% per year.
a. What is your estimate of the intrinsic value of a share of the stock?
b. If the market price of a share is equal to this intrinsic value, what is the expected dividend yield?
c. What do you expect its price to be one year from now? Is the implied capital gain consistent with your
estimate of the dividend yield and the market capitalization rate?
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SOLUTION:
Year
Dividends per share
Growth rate
0
1
1
1.25
.25
2
1.5625
.25
3
1.953125
.25
4
2.05078
.05
a. 2-Stage starts the constant growth stage in year 4. Therefore:
P3 = D4/(k – g) = 2.05078/(.20 -.05) = $13.67
P0 = D1/(1+k) + D2/(1+k)2 + (D3 + P3)/(1+k)3 = $1.25/1.2 + $1.5625/1.22 + ($1.953 + $13.67)/1.23 = $11.17
b. If the market price of a share is equal to this intrinsic value, the expected dividend yield is D1/P0, which is
1.25/11.17 = .1119 or 11.2%
c. Its price one year from now = P1 = D2/(1+k) + (D3 + P3)/(1+k)2 = $1.5625/1.2 + ($1.953 + $13.67)/1.2 2 =
$12.15.
The implied capital gain is $12.15 - $11.17 = $.98, which is 8.8% of the price P 0. Thus the dividend yield plus
the capital gain rate add up to 20%, which is k.
8. The Bearded ladies’ Stock guide offers the following method for selecting stocks:
Compute the stock’s PEG ratio by dividing its P/E multiple by its growth rate of earnings. Select only those
stocks whose PEG ratio is in the lowest quartile.
a. If the stock is fairly priced according to the constant-growth-rate DDM, what should be its PEG ratio as
a function of the following three variables: the stock’s market capitalization rate (k), the expected
profitability of its future investments (ROE), and its plowback ratio (b)? (Assume the P/E ratio used in
computing PEG is the ratio of the stock’s current price to its expected earnings per share, P 0/E1)
b. Assume the CAPM and the DDM are valid. The risk free rate is .04 and the risk premium on the market
portfolio is .06. What should be the relationship between the PEG for a stock whose ROE is .10 and a
stock whose ROE is .15, assuming the two stocks have the same beta (equal to 1) and plowback ratio
(equal to .6)?
c. What do you think of the Bearded Ladies’ method?
SOLUTION:
a. If the DDM holds we know that P0 = D1 / (k-g), furthermore, we know that g = b x ROE and D1=(1-b) E1
P0
(1  b)E1
1

E
k b(ROE) E1
1 b
PEG  1  

(ROE)
b
g
b(ROE)(kb(ROE))
b.
The values of k for each of the stocks will be: k = .04 + .06 beta = .04 + .06 =.1
PEG1= (1-.6) / (.6 x .10 x (.1 - .6 x .10)) = 166.67
PEG2= (1-.6) / (.6 x .15 x (.1 - .6 x .15)) = 444.44
PEG1 < PEG2
c.
As we can see in part b, the PEG rule would lead us to choose the stock with the lowest ROE.
In general, if the stock market is informally efficient, then any stock will offer an expected rate of return that is
commensurate with the stock’s perceived market risk, regardless of the stock’s PEG.
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Using the Internet for Stock Pricing
9. Pick a company whose stock is traded on the NYSE. Use one of the stock valuation models discussed in
this chapter together with information that you can find by searching the Internet to compute an intrinsic
value for the stock. Compare your estimate of intrinsic value with the stock’s actual price. Would you be
willing to make an investment decision on the basis of your research? Why or why not?
SOLUTION:
One simple model that we can use to value a company is to find the average P/E multiple of the industry in which the
company operates and multiply it by the expected earnings per share of that company. The difference between this
intrinsic value and the actual market value of the stock can be explained by the difference between our assumptions
regarding the company’s future investment opportunities and the market’s expectations. For example, if the market
value of the stock is higher than the intrinsic value found, then this difference reflects the investors’ belief that the
company will have a greater-than-average future investments opportunities with a rate of return greater than the
market capitalization rate for this particular industry. If markets are efficient, then this market value is supposed to be
the “real” value of the company, and represents the view of the majority of investors, hence I would be reluctant to
follow my own findings of the intrinsic value as a basis for an investment decision.
Dividend Policy
10. Divido Corporation is an all-equity financed firm with a total market value of $100 million. The company
holds $10 million in cash-equivalents and has $90 million in other assets. There are 1,000,000 shares of Divido
common stock outstanding, each with a market price of $100. What would be the impact on Divido’s stock
price and on the wealth of its shareholders of each of the following decisions? Consider each decision
separately.
a. The company pays a cash dividend of $10 per share.
b. The company repurchases 100,000 shares.
c. The company pays a 10% stock dividend.
d. The company has a 2-for-1 stock split.
e. The company invests $10 million in an expansion that has an expected IRR equal to the firm’s cost of
capital.
SOLUTION:
a. The stock price falls by $10, but shareholder wealth remains the same in a frictionless world because
shareholders receive $10 in cash on each share they own. In the real world, shareholder’s wealth may decline
because personal taxes may have to be paid on the cash dividend.
b. The stock price is unchanged and so is shareholder wealth. Some of the shareholders who sold their shares may
have to pay taxes on their capital gains in the real world.
c. The number of shares outstanding rises to 1,100,000, and the stock price falls to $90.909 (=$100MM/$1.1MM)
per share. Shareholder wealth is unchanged: instead of having one share at $100, now the shareholder will have
1.1 shares at $90.909/share (1.1 x 90.909 = 100)
d. The number of shares outstanding rises to 2,000,000, and the stock price falls to $50 per share. Theoretically,
shareholder wealth is unchanged.
e. The composition of the firm’s assets changes. Cash falls by $10 million and other assets go up by the same
amount. There is no change in either the stock price or in shareholder wealth.
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11. It has been found empirically, that on average the total market value of their stock rises when firms
announce a stock split. What hypotheses might you offer to explain this phenomenon?
SOLUTION:
Theoretically, when a firm announces a stock-split, the number of shares doubles (if 2-to-1 stock split) and the
market value per share drops by half. Empirically, we have observed a small increase in market value of the stock
after the announcement of a stock-split. This can be explained by the informational content of the split. Outside
investors may interpret this stock dividend as a positive sign that the company is doing well, hence increasing the
price of the stock. Another possible interpretation is that since the price per share is now lower after the split, it can
become more affordable for some investors.
12. Suppose that a company has had an extraordinarily profitable year, and it announces that it will use most
of its net cash inflow to buy back shares of its stock in the market. Would you expect the price of its stock to
rise or fall when the announcement is made? Explain.
SOLUTION:
Theoretically, the price of the stock should not change after a stock repurchase. But the announcement could send a
positive signal to investors that the company has been doing very well and has enough cash to buy back shares as a
form of dividends. This might increase the price of the stock after the announcement.
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