CHAPTER 9 –
1
CHAPTER 9
STOCK VALUATION
Answers to Concept Questions
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
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
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. 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.
10. 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.
CHAPTER 9 –
2
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) = $2.07(1.043)/(.11 – .043)
P0 = $32.22
The dividend at Year 4 is the dividend today times the FVIF for the growth rate in dividends and 4
years, so:
There is another feature of the constant dividend growth model: The stock price grows at the dividend
growth rate. 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 3 years, and
we have already calculated the stock price today. The stock price in 3 years will be:
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:
3. The dividend yield is the dividend next year divided by the current price, so the dividend yield is:
Dividend yield = D1/P0
4. Using the constant growth model, we find the price of the stock today is:
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:
6. We know the stock has a required return of 9.9 percent, and the dividend and capital gains yield are
equal, so:
Dividend yield = 1/2(.099) = .0495 = Capital gains yield
Now we know both the dividend yield and capital gains yield. The dividend is the stock price times
7. The price of any financial instrument is the PV of the future cash flows. The future dividends of this
8. The price of 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 that
9. The growth rate of earnings is the return on equity times the retention ratio, so:
g = ROE × b
10. 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:
CHAPTER 9 –
5
Intermediate
11. 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:
12. Here we have a stock that pays no dividends for 9 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
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:
P9 = D10/(R – g)
13. 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:
14. With differential 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
differential growth period. The stock begins constant growth in Year 5, so we can find the price of the
15. With differential 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
differential growth period. The stock begins constant growth in Year 4, so we can find the price of the
16. 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(1.25)3
And the dividend in Year 4 will be the dividend in Year 3 times one plus the growth rate, or:
CHAPTER 9 –
7
When we solve this equation, we find that the stock price in Year 4 is 39.99 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:
17. 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:
18. 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:
19. The price of a share of preferred stock is the dividend payment divided by the required return. We
know the dividend payment in Year 5, so we can find the price of the stock in Year 4, one year before
the first dividend payment. Doing so, we get:
20. The dividend yield is the annual dividend divided by the stock price, so:
Dividend yield = Dividend/Stock price
.019 = Dividend/$45.18
23. Since we know the stock price as well, we can use the PE ratio to solve for EPS as follows:
PE = Stock price/EPS
$1.964 = Net income/30,000,000
21. Here we have a stock paying a constant dividend for a fixed period, and an increasing dividend
thereafter. We need to find the present value of the two different cash flows using the appropriate
quarterly interest rate. The constant dividend is an annuity, so the present value of these dividends is:
PVA = C(PVIFAR,t)
22. Here we need to find the dividend next year for a stock with irregular growth in dividends. We know
the stock price, the dividend growth rate, and the required return, but not the dividend. First, we need
to realize that the dividend in Year 3 is the constant dividend times the FVIF. The dividend in Year 3
will be:
D3 = D(1.035)
The equation for the stock price will be the present value of the constant dividends, plus the present
value of the future stock price, or:
23. The required return of a stock consists of two components, the capital gains yield and the dividend
yield. In the constant dividend growth model (growing perpetuity equation), the capital gains yield is
the same as the dividend growth rate, or algebraically:
R = D1/P0 + g
We can find the dividend growth rate by the growth rate equation, or:
24. First, we need to find the annual dividend growth rate over the past four years. To do this, we can use
the future value of a lump sum equation, and solve for the interest rate. Doing so, we find the dividend
growth rate over the past four years was:
25. a. We can find the price of all the outstanding company stock by using the dividends the same way
we would value an individual share. Since earnings are equal to dividends, and there is no growth,
the value of the company’s stock today is the present value of a perpetuity, so:
PE = 9.09 times
b. Since the earnings have increased, the price of the stock will increase. The new price of the
outstanding company stock is:
PE = 10.98 times
CHAPTER 9 –
11
c. Since the earnings have increased, the price of the stock will increase. The new price of the
PE = 12.88 times
26. a. 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 19, we find:
P = 19($3.47)
P = $65.93
27. We need to find the enterprise value of the company. We can calculate EBITDA as sales minus costs,
so:
EBITDA = Sales – Costs
EBITDA = $48,000,000 – 15,000,000
EBITDA = $33,000,000
28. a. To value the stock today, we first need to calculate the cash flows for the next 6 years. The sales,
costs, and net investment all grow by the same rate, namely 14 percent, 12 percent, 10 percent,
and 8 percent, respectively, for the following 4 years, then 6 percent indefinitely. So, the cash
flows for each year will be:
Year 2
Year 3
Year 4
Year 5
Year 6
Sales
$131,100,000
$146,832,000
$161,515,200
$174,436,416
$184,902,601
Costs
76,380,000
85,545,600
94,100,160
101,628,173
107,725,863
EBT
Taxes
Net income
$48,416,256
$53,257,882
$57,518,512
$60,969,623
Investment
13,680,000
15,321,600
16,853,760
Cash flow
$33,094,656
$36,404,122
$39,316,451
$41,675,438
To find the terminal value of the company in Year 6, we can discount the Year 7 cash flows as a
growing perpetuity, which will be:
b. In this case, we are going to use the PE multiple to find the terminal value. All of the cash flows
from part a will remain the same. So, the terminal value in Year 6 is:
Terminal value = 11($60,969,623)
Terminal value = $670,665,851
Under this assumption for the terminal value, the value of the company today is:
29. We are asked to find the dividend yield and capital gains yield for each of the stocks. All of the stocks
have a 15 percent required return, which is the sum of the dividend yield and the capital gains yield.
To find the components of the total return, we need to find the stock price for each stock. Using this
stock price and the dividend, we can calculate the dividend yield. The capital gains yield for the stock
will be the total return (required return) minus the dividend yield.
30. a. Using the constant growth model, the price of the stock paying annual dividends will be:
P0 = D0(1 + g)/(R – g)
P0 = $3.80(1.045)/(.11 – .045)
P0 = $61.09
b. If the company pays quarterly dividends instead of annual dividends, the quarterly dividend
31. Here we have a stock with differential growth, but the dividend growth changes every year for the first
four years. We can find the price of the stock in Year 3 since the dividend growth rate is constant after
the third dividend. The price of the stock in Year 3 will be the dividend in Year 4, divided by the
32. Here we want to find the required return that makes the PV of the dividends equal to the current stock
price. The equation for the stock price is: