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CHAPTER 8
STOCK VALUATION
Answers to Concepts Review and Critical Thinking Questions
1. The value of any investment depends on the present value of its cash flows; i.e., what investors will
2. Investors believe the company will eventually start paying dividends (or be sold to another
3. In general, companies that need the cash will often forgo dividends since dividends are a cash
4. 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 (a) if dividends are
expected to occur forever, that is, the stock provides dividends in perpetuity, and (b) if a constant
5. The common stock probably has a higher price because the dividend can grow, whereas it is fixed on
6. The two components are the dividend yield and the capital gains yield. For most companies, the
8. In a corporate election, you can buy votes (by buying shares), so money can be used to influence or
9. It wouldn’t seem to be unfair or unethical. Investors who don’t like the voting features of a particular
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10. Investors buy such stock because they want it, recognizing that the shares have no voting power.
11. Presumably, the current stock value reflects the risk, timing and magnitude of all future cash flows,
12. If this assumption is violated, the two-stage dividend growth model is not valid. In other words, the
13. In a declassified board, every board seat is up for election every year. This structure allows investors
to vote out a director (and even the entire board) much more quickly if investors are dissatisfied.
14. The major difficulty in using price ratio analysis is determining the correct benchmark PE ratio. In a
previous chapter, we showed how the sustainable growth rate is determined, and in a future chapter
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. The constant dividend growth model is:
So the price of the stock today is:
P0 = D0(1 + g)/(R – g)
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P3 = D3(1 + g)/(R – g)
We can do the same thing to find the dividend in Year 16, which gives us the price in Year 15, so:
P15 = D15(1 + g)/(R – g)
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
P3 = P0(1 + g)3
And the stock price in 15 years will be:
P15 = P0(1 + g)15
2. 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 = (D1/P0) + g
3. The dividend yield is the dividend next year divided by the current price, so the dividend yield is:
Dividend yield = D1/P0
The capital gains yield, or percentage increase in the stock price, is the same as the dividend growth
rate, so:
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4. Using the constant growth model, we find the price of the stock today is:
P0 = D1/(R – g)
5. 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
6. We know the stock has a required return of 10.6 percent, and the dividend and capital gains yields
are equal, so:
Now we know both the dividend yield and capital gains yield. The dividend is the stock price times
the dividend yield, so:
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:
We can solve for the dividend that was just paid:
7. The price of any financial instrument is the PV of the future cash flows. The future dividends of this
stock are an annuity for 13 years, so the price of the stock is the PVA, which will be:
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9. We can use the constant dividend growth model, which is:
Pt = Dt × (1 + g)/(R – g)
So the price of each company’s stock today is:
Red stock price = $3.65/(.08 – .04) = $91.25
As the required return increases, the stock price decreases. This is a function of the time value of
10. If the company uses straight voting, you will need to own one-half of the shares, plus one share, in
order to guarantee enough votes to win the election. So, the number of shares needed to guarantee
election under straight voting will be:
And the total cost to you will be the shares needed times the price per share, or:
11. If the company uses cumulative voting, you will need 1/(N + 1) percent of the stock (plus one share)
to guarantee election, where N is the number of seats up for election. So, the percentage of the
company’s stock you need will be:
Percent of stock needed = 1/(N + 1)
So, the number of shares you need to purchase is:
And the total cost to you will be the shares needed times the price per share, or:
12. Using the equation to calculate the price of a share of stock with the PE ratio:
P = Benchmark PE ratio × EPS
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With a PE ratio of 18, we find:
And with a PE ratio of 21, we find:
13. First, we need to find the sales per share, which is:
Sales per share = Sales/Shares outstanding
Using the equation to calculate the price of a share of stock with the PS ratio:
P = Benchmark PS ratio × Sales per share
With a PS ratio of 4.3, we find:
And with a PS ratio of 3.6, we find:
Now we can find the price of the stock in Year 3. We need to find the price in Year 3 since the
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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:
15. 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 the general constant dividend growth formula is:
This means that since we will use the dividend in Year 10, we will be finding the stock price in Year
The price of the stock today is the PV of the stock price in the future. We discount the future stock
price at the required return. The price of the stock today will be:
16. The price of a stock is the PV of the future dividends. This stock is paying five dividends, so the
price of the stock is the PV of these dividends using the required return. The price of the stock is:
17. With nonconstant 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
P4 = D4 (1 + g)/(R – g)
The price of the stock today is the PV of the first four dividends, plus the PV of the Year 4 stock
price. So, the price of the stock today will be:
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18. 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 3, one year before the constant dividend growth begins, as:
P3 = D3(1 + g)/(R – g)
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:
We could also use the two-stage dividend growth model for this problem, which is:
P0 = [D0(1 + g1)/(R – g1)]{1 – [(1 + g1)/(1 + R)]t}+ [(1 + g1)/(1 + R)]t[D0(1 + g2)/(R – g2)]
19. Here we need to find the dividend next year for a stock experiencing supernormal growth. We know
the stock price, the dividend growth rates, and the required return, but not the dividend. First, we
And the dividend in Year 4 will be the dividend in Year 3 times one plus the growth rate, or:
The stock begins constant growth in Year 4, so we can find the price of the stock in Year 4 as the
Now we can substitute the previous dividend in Year 4 into this equation as follows:
P4 = D0(1 + g1)3(1 + g2)(1 + g3)/(R – g)
When we solve this equation, we find that the stock price in Year 4 is 59.52 times as large as the
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We can factor out D0 in the equation and combine the last two terms. Doing so, we get:
Reducing the equation even further by solving all of the terms in the braces, we get:
This is the dividend today, so the projected dividend for the next year will be:
21. We are given the stock price, the dividend growth rate, and the required return and are asked to find
the dividend. Using the constant dividend growth model, we get:
Solving this equation for the dividend gives us:
22. The price of a share of preferred stock is the dividend payment divided by the required return. We
know the dividend payment in Year 20, so we can find the price of the stock in Year 19, one year
before the first dividend payment. Doing so, we get:
The price of the stock today is the PV of the stock price in the future, so the price today will be:
23. The annual dividend paid to stockholders is $1.75, and the dividend yield is 2.35 percent. Using the
equation for the dividend yield:
Dividend yield = Dividend/Stock price
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We can plug the numbers in and solve for the stock price:
.0235 = $1.75/P0
The “Net Chg” of the stock shows the stock decreased by $.23 on this day, so the closing stock price
yesterday was:
To find the net income, we need to find the EPS. The stock quote tells us the PE ratio for the stock is
19. Since we know the stock price as well, we can use the PE ratio to solve for EPS as follows:
PE = Stock price/EPS
We know that EPS is just the total net income divided by the number of shares outstanding, so:
EPS = Net income/Shares
