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CHAPTER 6
STOCK VALUATION
Answers to Concept 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 company).
3. 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. 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 applying the general method of valuation explained in this chapter. A
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.
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.
rate and the capital gains yield are the same.
8. The three factors are: 1) The company’s future growth opportunities. 2) The company’s level of risk,
which determines the interest rate used to discount cash flows. 3) The accounting method used.
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.
under no obligation to buy it.
11. 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.
12. Presumably, the current stock value reflects the risk, timing and magnitude of all future cash flows,
both short-term and long-term. If this is correct, then the statement is false.
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:
Pt = Dt × (1 + g)/(R – g)
So, the price of the stock today is:
P0 = D0(1 + g)/(R – g)
The dividend at Year 4 is the dividend today times the FVIF for the growth rate in dividends and four
years, so:
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 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:
P3 = P0(1 + g)3
P3 = $38.18(1 + .04)3
P15 = P0(1 + g)15
P15 = $38.18(1 + .04)15
equation for R. Doing so, we find:
R = (D1/P0) + g
R = ($2.14/$32) + .044
3. The dividend yield is the dividend next year divided by the current price, so the dividend yield is:
Dividend yield = D1/P0
Dividend yield = $2.14/$32
Dividend yield = .0669, or 6.69%
P0 = D1/(R – g)
P0 = $3.08/(.11 – .046)
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
R = .041 + .054
6. We know the stock has a required return of 12 percent, and the dividend and capital gains yield are
equal, so:
Dividend yield = 1/2(.104) = .052 = Capital gains yield
Now we know both the dividend yield and capital gains yield. The dividend is the stock price times
the dividend yield, so:
D1 = .052($63)
D1 = $3.28
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:
D1 = D0(1 + g)
We can solve for the dividend that was just paid:
$3.28 = D0(1 + .052)
D0 = $3.28/1.052
D0 = $3.11
stock are an annuity for 8 years, so the price of the stock is the PVA, which will be:
P0 = $10.25(PVIFA9.7%,8)
P0 = $55.29
equation as the constant growth model, with a dividend growth rate of zero percent. Remember that
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/P0
R = $3.85/$108
R = .0356, or 3.56%
g = ROE × b
g = .14(.75)
g = .1050, or 10.50%
Next year’s earnings = Current earnings(1 + g)
Next year’s earnings = $38,600,000(1 + .1050)
Next year’s earnings = $42,653,000
Pt = Dt(1 + g)/(R – g)
So the price of each company’s stock today is:
Red stock price = $2.86/(.08 – .05) = $95.33
Yellow stock price = $2.86/(.11 – .05) = $47.67
Blue stock price = $2.86/(.14 – .05) = $31.78
money: A higher discount rate decreases the present value of cash flows. It is also important to note
that relatively small changes in the required return can have a dramatic impact on the stock price.
11. 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:
Shares needed = (415,000 shares/2) + 1
12. 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)
Percent of stock needed = 1/(4 + 1)
Percent of stock needed = .20, or 20%
So, the number of shares you need to purchase is:
Number of shares to purchase = (415,000 × .20) + 1
13. Using the equation to calculate the price of a share of stock with the PE ratio:
P = Benchmark PE ratio × EPS
So, with a PE ratio of 18, we find:
P = 18($2.14)
And with a PE ratio of 21, we find:
P = 21($2.14)
Intermediate
14. 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
be the dividend in Year 7, divided by the required return minus the growth rate in dividends. So:
P6 = D6 (1 + g)/(R – g)
P6 = D0 (1 + g)7/(R – g)
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:
P3
= $2.73(1.04)4/1.13 + $2.73(1.04)5/1.132 + $2.73(1.04)6/1.133 + $59.87/1.133
P3 = $49.32
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:
P0 = $2.73(1.04)/1.15 + $2.73(1.04)2/(1.15)2 + $2.73(1.04)3/(1.15)3 + $49.32/(1.15)3
P0 = $39.15
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 form of the constant dividend growth formula is:
Pt = [Dt × (1 + g)]/(R – g)
This means that since we will use the dividend in Year 12, we will be finding the stock price in Year
11. 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 11 will be:
P11 = D12/(R – g)
P11 = $17/(.13 – .055)
of the stock is the PV of these dividends using the required return. The price of the stock is:
P0 = $16.50/1.11 + $20/1.112 + $23.50/1.113 + $27/1.114 + $30.50/1.115
P0 = $84.17
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 5, so we can find the price of
the stock in Year 4, one year before the constant dividend growth begins, as:
P4 = D4(1 + g)/(R – g)
P4 = $2.85(1.05)/(.10 – .05)
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)
P3 = D0(1 + g1)3(1 + g2)/(R – g2)
price. The price of the stock today will be:
P0 = $2.65(1.27)/1.104 + $2.65(1.27)2/1.1042 + $2.65(1.27)3/1.1043 + $96.14/1.1043
P0 = $82.04
19. Here we need to find the dividend next year for a stock experiencing differential growth. We know the
stock price, the dividend growth rates, and the required return, but not the dividend. First, we need to
realize that the dividend in Year 3 is the current dividend times the FVIF. The dividend in Year 3 will
be:
D3 = D0(1.27)3
When we solve this equation, we find that the stock price in Year 4 is 41.90 times as large as the
dividend today. Now we need to find the equation for the stock price today. The stock price today is
the PV of the dividends in Years 1, 2, 3, and 4, plus the PV of the Year 4 price. So:
20. The constant growth model can be applied even if the dividends are declining by a constant percentage,
just make sure to recognize the negative growth. So, the price of the stock today 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:
D0 = $3.26
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 10, so we can find the price of the stock in Year 9, one year before
the first dividend payment. Doing so, we get:
