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Chapter 8

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CHAPTER 8
STOCK VALUATION
Learning Objectives
LO1 How stock prices depend on future dividends and dividend growth.
LO2 The characteristics of common and preferred stocks.
LO3 The different ways corporate directors are elected to office.
LO4: The stock market quotations and the basics of stock market reporting.
Answers to Concepts Review and Critical Thinking Questions
1. (LO1) The value of any investment depends on its cash flows; i.e., what investors will actually receive. The
cash flows from a share of stock are the dividends.
2. (LO1) Investors believe the company will eventually start paying dividends (or be sold to another company).
3. (LO1) In general, companies that need the cash will often forgo dividends since dividends are a cash expense.
Young, growing companies with profitable investment opportunities are one example; another example is a
company in financial distress. This question is examined in depth in a later chapter.
4. (LO1) The general method for valuing a share of stock is to find the present value of all expected future
dividends. The dividend growth model presented in the text is only valid (i) if dividends are expected to occur
forever, that is, the stock provides dividends in perpetuity, and (ii) if a constant growth rate of dividends occurs
forever. A violation of the first assumption might be a company that is expected to cease operations and
dissolve itself some finite number of years from now. The stock of such a company would be valued by the
methods of this chapter by applying the general method of valuation. A violation of the second assumption
might be a start-up firm that isn’t currently paying any dividends, but is expected to eventually start making
dividend payments some number of years from now. This stock would also be valued by the general dividend
valuation method of this chapter.
5. (LO2) The common stock probably has a higher price because the dividend can grow, whereas it is fixed on
the preferred. However, the preferred is less risky because of the dividend and liquidation preference, so it is
possible the preferred could be worth more, depending on the circumstances.
6. (LO1) The two components are the dividend yield and the capital gains yield. For most companies, the capital
gains yield is larger. This is easy to see for companies that pay no dividends. For companies that do pay
dividends, the dividend yields are rarely over five percent and are often much less.
7. (LO1) Yes. If the dividend grows at a steady rate, so does the stock price. In other words, the dividend growth
rate and the capital gains yield are the same.
8. (LO3) In a corporate election, you can buy votes (by buying shares), so money can be used to influence or
even determine the outcome. Many would argue the same is true in political elections, but, in principle at least,
no one has more than one vote.
9. (LO3) It wouldn’t seem to be. Investors who don’t like the voting features of a particular class of stock are
under no obligation to buy it.
10. (LO2) Investors buy such stock because they want it, recognizing that the shares have no voting power.
Presumably, investors pay a little less for such shares than they would otherwise.
S8-1 Solutions to Questions and Problems
NOTE: All end of chapter problems were solved using a spreadsheet. Many problems require multiple steps. Due to
space and readability constraints, when these intermediate steps are included in this solutions manual, rounding
may appear to have occurred. However, the final answer for each problem is found without rounding during any
step in the problem.
Basic
1. (LO1) The constant dividend growth model is:
P = D × (1 + g) / (R – g)
t t
So the price of the stock today is:
P 0 D (10+ g) / (R – g) = $1.95 (1.06) / (.11 – .06) = $41.34
The dividend at year 4 is the dividend today times the FVIF for the growth rate in dividends and four years, so:
4 4
P 3 D (13+ g) / (R – g) = D (1 +0g) / (R – g) = $1.95 (1.06) / (.11 – .06) = $49.24
We can do the same thing to find the dividend in Year 16, which gives us the price in Year 15, so:
16 16
P 15= D 15 (1 + g) / (R – g) = D 01 + g) / (R – g) = $1.95 (1.06) / (.11 – .06) = $99.07
There is another feature of the constant dividend growth model: The stock price grows at the dividend growth
rate. So, if we know the stock price today, we can find the future value for any time in the future we want to
calculate the stock price. In this problem, we want to know the stock price in three years, and we have already
calculated the stock price today. The stock price in three years will be:
3 3
P 3 P (0 + g) = $41.34(1 + .06) = $49.24
And the stock price in 15 years will be:
15 15
P 15= P 01 + g) = $41.34(1 + .06) = $99.07
2. (LO1) We need to find the required return of the stock. Using the constant growth model, we can solve the
equation for R. Doing so, we find:
R = (D /1P )0+ g = ($2.10 / $48.00) + .05 = .09375or 9.375%
3. (LO1) The dividend yield is the dividend next year divided by the current price, so the dividend yield is:
Dividend yield = D / P1= $0.10 / $48.00 = .04375 or 4.375%
The capital gains yield, or percentage increase in the stock price, is the same as the dividend growth rate, so:
Capital gains yield = 5%
4. (LO1) Using the constant growth model, we find the price of the stock today is:
P = D / (R – g) = $3.04 / (.11 – .038) = $42.22
0 1
S8-2 5. (LO1) The required return of a stock is made up of two parts: The dividend yield and the capital gains yield.
So, the required return of this stock is:
R = Dividend yield + Capital gains yield = .063 + .052 = .115 or 11.5%
6. (LO1) We know the stock has a required return of 11 percent, and the dividend and capital gains yield are
equal, so:
Dividend yield = 1/2(.11) = .055 = Capital gains yield
Now we know both the dividend yield and capital gains yield. The dividend is simply the stock price times the
dividend yield, so:
D = .055($47) = $2.585
1
This is the dividend next year. The question asks for the dividend this year. Using the relationship between the
dividend this year and the dividend next year:
D 1 D (0 + g)
We can solve for the dividend that was just paid:
$2.585 = D (0 + .055)
D = $2.585 / 1.055 = $2.45
0
7. (LO1) The price of any financial instrument is the PV of the future cash flows. The future dividends of this
stock are an annuity for eight years, so the price of the stock is the PVA, which will be:
P0= $9.75(PVIFA 10%,11 = $63.33
8. (LO1) The price a share of preferred stock is the dividend divided by the required return. This is the same
equation as the constant growth model, with a dividend growth rate of zero percent. Remember, most preferred
stock pays a fixed dividend, so the growth rate is zero. Using this equation, we find the price per share of the
preferred stock is:
R = D/P 0 $5.50/$108 = .0509 or 5.09%
S8-3 Intermediate
9. (LO1) This stock has a constant growth rate of dividends, but the required return changes twice. To find the
value of the stock today, we will begin by finding the price of the stock at Year 6, when both the dividend
growth rate and the required return are stable forever. The price of the stock in Year 6 will be the dividend in
Year 7, divided by the required return minus the growth rate in dividends. So:
7 7
P 6 D (16+ g) / (R – g) = D (1 +0g) / (R – g) = $3.50 (1.05) / (.10 – .05) = $98.50
Now we can find the price of the stock in Year 3. We need to find the price here since the required return
changes at that time. The price of the stock in Year 3 is the PV of the dividends in Years 4, 5, and 6, plus the
PV of the stock price in Year 6. The price of the stock in Year 3 is:
4 5 2 6 3 3
P 3 $3.50(1.050) / 1.12 + $3.50(1.050) / 1.12 + $3.50(1.05) / 1.12 + $98.50 / 1.12
P = $80.80
3
Finally, we can find the price of the stock today. The price today will be the PV of the dividends in Years 1, 2,
and 3, plus the PV of the stock in Year 3. The price of the stock today is:
2 2 3 3 3
P 0 $3.50(1.050) / 1.14 + $3.50(1.050) / (1.14) + $3.50(1.050) / (1.14) + $80.80 / (1.14)
P = $63.46
0
10. (LO1) Here we have a stock that pays no dividends for 10 years. Once the stock begins paying dividends, it
will have a constant growth rate of dividends. We can use the constant growth model at that point. It is
important to remember that general constant dividend growth formula is:
P = [D × (1 + g)] / (R – g)
t t
This means that since we will use the dividend in Year 10, we will be finding the stock price in Year 9. The
dividend growth model is similar to the PVA and the PV of a perpetuity: The equation gives you the PV one
period before the first payment. So, the price of the stock in Year 9 will be:
P = D / (R – g) = $10.00 / (.14 – .05) = $111.11
9 10
The price of the stock today is simply the PV of the stock price in the future. We simply discount the future
stock price at the required return. The price of the stock today will be:
9
P 0 $111.11 / 1.14 = $34.17
11. (LO1) The price of a stock is the PV of the future dividends. This stock is paying four dividends, so the price
of the stock is the PV of these dividends using the required return. The price of the stock is:
P = $10 / 1.11 + $14 / 1.11 + $18 / 1.11 + $22 / 1.11 = $48.034
0
12. (LO1) With supernormal dividends, we find the price of the stock when the dividends level off at a constant
growth rate, and then find the PV of the future stock price, plus the PV of all dividends during the supernormal
growth period. The stock begins constant growth in Year 4, so we can find the price of the stock in Year 4, at
the beginning of the constant dividend growth, as:
P 4 D (14+ g) / (R – g) = $2.00(1.05) / (.12 – .05) = $30.00
The price of the stock today is the PV of the first four dividends, plus the PV of the Year 3 stock price. So, the
price of the stock today will be:
P = $11.00 / 1.12 + $8.00 / 1.12 + $5.00 / 1.12 + $2.00 / 1.12 + $30.00 / 1.12 = $40.09 4
0
S8-4 13. (LO1) With supernormal dividends, we find the price of the stock when the dividends level off at a constant
growth rate, and then find the PV of the futures stock price, plus the PV of all dividends during the
supernormal growth period. The stock begins constant growth in Year 4, so we can find the price of the stock
in Year 3, one year before the constant dividend growth begins as:
3
P 3 D (1 3 g) / (R – g) = D (1 + g0) (1 + g1) / (R – g2
P 3 $1.80(1.30) (1.06) / (.13 – .06)
P 3 $59.88
The price of the stock today is the PV of the first three dividends, plus the PV of the Year 3 stock price. The
price of the stock today will be:
2 2 3 3 3
P 0 1.80(1.30) / 1.13 + $1.80(1.30) / 1.13 + $1.80(1.30) / 1.13 + $59.88 / 1.13
P 0 $48.69
We could also use the two-stage dividend growth model for this problem, which is:
T T
P 0 [D (10+ g )/(1 – g )]{11– [(1 + g )/(1 +1R)] }+ [(1 + g )/(1 + R)1 [D (1 + g )/(R0– g )] 1 1
3

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